User talk:PrimeHunter/Archive 4

Brun's constant
Stop reverting my edits. I'm warning you. This is about math, and not about your belief.Eganfan (talk) 21:53, 4 July 2010 (UTC)


 * My edits are based on a cited reliable source, not on my belief. Your unsourced edit warring is apparently based on your own belief that the reliable source is wrong. That's not how Verifiability works. PrimeHunter (talk) 23:52, 4 July 2010 (UTC)
 * There is no source that shows the proof. Eganfan (talk) 00:08, 5 July 2010 (UTC)
 * Verifiability doesn't require that. Wikipedia is not a math journal. It's an encyclopedia based on published reliable sources. You apparently think Crandall & Pomerance are wrong but they weigh more than an anonymous Wikipedia editor here. PrimeHunter (talk) 00:18, 5 July 2010 (UTC)
 * Have you ever read their book? Give me one reason you think this result is verifiable.Eganfan (talk) 09:06, 5 July 2010 (UTC)
 * I have read the part stating this result. As I keep saying, it satisfies Verifiability. Have you read that policy? Give me one reason why you think this result is not verifiable as defined by the Wikipedia policy. PrimeHunter (talk) 11:12, 5 July 2010 (UTC)
 * Because there is no proof other than they say they have. This is like Fermat stating he has the proof of FLT. You're the one who should read Verifiability. Eganfan (talk) 13:09, 5 July 2010 (UTC)
 * Fermat's claim was written in a book margin. He didn't publish it and it was an unreliable source. It was nothing like Crandall & Pomerance and wouldn't satisfy Verifiability. I don't see a part of that policy which disallows to cite a published and respected book by Crandall & Pomerance. PrimeHunter (talk) 13:29, 5 July 2010 (UTC)
 * Any respected book could have a mistake.Eganfan (talk) 13:37, 5 July 2010 (UTC)
 * Yes, any source considered reliable by Wikipedia could have a mistake, and some of them do, but Wikipedia has decided to rely on information in reliable sources for its millions of articles. An alleged simple short proof of FLT would fall under Verifiability, but an upper bound on B_2 does not. Crandall & Pomerance is OK for that purpose when there is no demonstrated reason to believe they are wrong. Other mathematicians have also reported their result without reservations. PrimeHunter (talk) 18:36, 5 July 2010 (UTC)
 * Well, the article is fixed and I'm quite satisfied with that.Eganfan (talk) 19:06, 5 July 2010 (UTC)

Novak Djokovic
Is there a way to mass revert PJAlvarez's edits - because it is tedious to go through changing them manually. Alan16 (talk) 13:28, 6 October 2009 (UTC)


 * I'll stop where I am. If you are using their contribution list to do it, I got down to Djokovic-Nadal rivalry. Alan16 (talk) 13:36, 6 October 2009 (UTC)

Category merge
I cannot delete Category:Barcelona KIA as you requested, because the speedy deletion criteria specifically state that this process does not apply to categories that have been "emptied out of process." If you want to merge or delete a category, you need to follow the process outlined at WP:CFD, and I invite you to do so now for this category. --R'n'B (call me Russ) 19:47, 7 October 2009 (UTC)

Thank you
Thank you for the message on my userpage, and you are right, I do have unified login. I had not even heard of the Mirandeselanguage, so I have learnt something new. ACEOREVIVED (talk) 22:41, 12 October 2009 (UTC)

New Prime Discovered
82818079787776757473727170696867666564636261605958575655545352515049484746454443424140393837363534333231302928272625242322212019181716151413121110987654321 is new, I think, and it looks like the only one in the sequence (checked to 1000). Perhaps you'll want to look at general questions related to this. I also have found that the finite sequence which begins nicely with primes at 1000999, 1000999998997 and 1000999998997996995994993 has exactly one other (475 digit) prime member, corresponding to final digits 843.Julzes (talk) 03:41, 18 October 2009 (UTC)

Oh, hold up now! Applying the PNT, one should get close to a harmonic series for the number of expected primes of the type given. So, who knows what the next member of the sequence is after the first at 82? I'll have to pursue this question further.Julzes (talk) 03:59, 18 October 2009 (UTC)


 * 828180...321 is old. says: "The only prime in the first 30954 terms (Weisstein, Mar. 21, 2009) of this sequence is the 82nd term 828180...321". Concatenations of digits in integer sequences are sometimes called Smarandache sequences  or something with Smarandache for specific sequences like Smarandache–Wellin number. People have searched primes in many such sequences.  shows the first case of primes made by concatenating k increasing or decreasing consecutive integers without having to start or end at 1. In 2003 I found the smallest probable prime with 44 increasing integers. It starts at 10348-32 and has 15324 digits. You could try 22 decreasing integers, the only remaining case in that puzzle. PrimeForm/GW is the fastest program for this purpose. PrimeHunter (talk) 13:58, 18 October 2009 (UTC)


 * Update: Eric W. Weisstein has just announced in that the next probable prime is 37765 37764 37763 ... 5 4 3 2 1. PrimeHunter (talk) 17:03, 6 April 2010 (UTC)
 * 177719 digits! This little space is classic, with your 15324-digit thing mentioned above.
 * I will have to try that problem, since I just became aware of a pair of soon to be sisters-in-law who will become 22 on Sunday.Julzes (talk) 04:59, 10 April 2010 (UTC)
 * Cool! By the way, I don't think I mentioned this, but I ALMOST made a biggish name for myself in this general area.  You know the smallest prime factor of the 14th Fermat number?  I reasoned that someone had already searched beyond where I was going to go in my search, and just shut it down to work on other things.  49 digits!?!?!  AARGH!
 * On another note, I'm about to use this conversation as evidence of my verisimilitude in the claim that all of the base-ten coincidences I discovered were learned in a short period of time by one person without a lot of programming skill (and just for the curiosity of anyone interested in this non-process process I've been using). That's in a thread at http://www.rational-skepticism.org titled 'Fundamental Question to Supernaturalists'.  Related to this is something I asked Professor Honaker in an e-mail that I should ask you.  I know you're constantly busy, but I'm looking into the possibility of co-ordinated research into the degree to which mathematical coincidences might be or have been mediated rather than purely found through hard work on reasonably naturally chosen questions.  It is going to be very hard for me to do this on my own.  So, I'm throwing out the possibility that you might want to engage on the question or know some one or more people who would.  Good to hear from you, at any rate.  You wouldn't be closely related to Adolf Anderssen, the chess champion known for the 'immortal game' who shares my birthday (1818), would you?  Oh, perhaps not (2 s's vs. 1).Julzes (talk) 21:32, 9 April 2010 (UTC)


 * The F14 factor has a 49-digit k in k*2^16+1. The total size is 54 digits. Andersen is a common surname in Denmark and I'm not related to any of the famous name bearers. I haven't investigated how coincidental your discoveries are. I don't want to participate in the research you mention and I don't have suggestions for others. PrimeHunter (talk) 01:49, 11 April 2010 (UTC)


 * Okay. Thanks for pointing out the extra 5 digits.  I would have found that if I had just let it run, I think but have no way of knowing other than to waste cpu time.  No need to investigate mine; that other people seem to have them is what I was interested in.Julzes (talk) 07:00, 11 April 2010 (UTC)
 * Just in the interest of not being seen as keeping you deliberately in the dark on this, the first post of the link http://www.rational-skepticism.org/paranormal/list-3-coincidences-you-know-of-for-debunking-t4476.html is how I identify myself aside from my name, James Guthrie Merickel; and the long late post sequence in http://www.rational-skepticism.org/creationism/fundamental-question-to-all-supernaturalists-t3517-300.html contains everything I could immediately think of on base-10 coincidences and some other stuff I am in the process of adding today.Julzes (talk) 13:20, 11 April 2010 (UTC)

Okay. I sort of had my doubts about the originality of the discovery, but I thought I would have run across it already. Thanks for the references. and terminology. It's also good to be informed on primality testing. I have been using the applet you referenced for factorizations, but where I want just a test I will look into the alternatives.Julzes (talk) 19:56, 18 October 2009 (UTC)

Oh, I get it. The number of digits has to change for certain cases. [I have had to remove my skeptical remark.]Julzes (talk) 20:06, 18 October 2009 (UTC)

Apropos of the discovery that does seem original, I've been crudely using that applet to look at the question of these declining Smarandache sequences from the angle of determining which numbers begin more than one such prime, disregarding the specifics of the lengths for now. Up to 90, the only numbers (I think) that start more than one number are 10, 22, 46, 48, 55, 73 and 79, and 73 is the only one that starts three of them. It is beginning to look like my 1000 case may be somewhat extraordinary. Feel free to simply mention that I discovered it and carry the inquiry where your better skills may lead you. I propose that consideration be given to the sequence 4 (for the sole first of 43), 10 (for 109 and 10987), 73, 1000?, .... Is 1000 next as I propose? Is it feasible to find a fifth term? Is the sequence finite or infinite, and if infinite can an asymptotic formula be given? I would not be at all surprised if the first three terms for 1000 were extraordinary in their own right. Is 1000 the first number to give three primes with the minimum length? Well, I don't know if you are interested, but these are questions I find worth asking. Julzes (talk) 09:04, 19 October 2009 (UTC) Argh! I just discovered something that must be known: 10099, 100999897, 100999897969594939291, and 1009998...6261 are prime. So my proposed sequence would have fourth term 100, not 1000. Oh, well. It's still interesting.Julzes (talk) 09:38, 19 October 2009 (UTC)


 * I confirm that 73 is the first to start 3 primes. 100, 874 and 1000 are the first 3 to start 4 primes. There is no case of 5 below 1000 and finding the smallest 5 may be hard . But it was easy to find a probably non-minimal case of at least 5 by first testing the small concatenations of a few numbers starting at n and then skipping unpromising n which produce no or few small primes. n = 67414 gives at least 5 primes ending at 67413, 67411, 67351, 67329, 67263. If there are more primes then they have above 10000 digits but I don't want to search for them. PrimeHunter (talk) 23:52, 19 October 2009 (UTC)
 * There are also at least 5 primes starting at 286930, 442468, 603850. 961984 has at least 6, ending at 961983, 961981, 961953, 961939, 961909, 961489. PrimeHunter (talk) 01:30, 20 October 2009 (UTC)

Perhaps some of this is new: A) 102n+10n-1 is prime for n=1,2,3,5,6,7,9,13,26,42,153,188,204,282. B) x2n+xn-1 is prime for n=1, 2, 3 and 4 for x=2, 19, 44 and 45. A suggests the question of whether there are any other x for which there are six primes in the first seven or seven primes in the first 12, 11, 10 or 9 members of the sequence. As for B, it's not clear to me if there are any more such x. 44 and 45 consecutively is a nice arithmetical factoid. Is there any x for which one gets a prime for n=1 through 5? I have another easier question to settle related to the 1000 result. We have the sequence 4, 10, 1000 for the beginning of a sequence of numbers that start with each of the numbers and generate the shortest possible primes (I'm assuming your 874 result does not supplant 1000). What is the fourth term in this sequence?Julzes (talk) 01:24, 20 October 2009 (UTC)


 * A096594 is 1, 2, 3, 5, 6, 7, 9, 13, 26, 42, 153, 188, 204, 282, 699, 2886. x2n+xn-1 is prime for n=1, 2, 3 and 4 for x=2, 19, but not for n=4 with x=44 and 45. x2n+xn-1 is prime for n=1, 2, 3, 4, 5 for x = 460724, 841785, 1099276, 1582404, 3925484, ... PrimeHunter (talk) 02:11, 20 October 2009 (UTC)

OOPS! My factoid on x=44 and 45 is only good for n=1, 2 and 3. Certainly not as big a deal. Sorry about that. I do note that there are no primes for your five x's with n=6 to 12. Is there a six-out-of-the-first-seven case like for 10 to be found? I don't imagine one would have to go very far to find some value other than 10 which is good for 7 out of the first 12, but 7 out of 9 would be pretty unusual, and maybe 8 out of 13 is a little rare too. I imagine there is a finite limit for all of the first so many n's. Do you think that would simply be 5? Or am I way off base in even assuming it's finite? Also, does the data you've already given provide the next term of {4, 10, 1000,...}?Julzes (talk) 04:31, 20 October 2009 (UTC)

I just looked at that link you just gave, and I'm glad I stopped at 282, since it would have taken quite a long time to get the next term.Julzes (talk) 04:37, 20 October 2009 (UTC)


 * 33008304 gives 6 out of 7 for n = 1, 3, 4, 5, 6, 7. 26997935 gives 8 out of 15 for n = 1, 3, 4, 5, 8, 10, 13, 15. PrimeHunter (talk) 14:02, 20 October 2009 (UTC)

So, am I right that the NEXT value after 10 that gives six out of seven is 33008304 and that you have no reason to doubt that 10 is unique for seven out of nine? By the way, the next n after 7 for 33008304 is greater than 300.Julzes (talk) 20:07, 20 October 2009 (UTC)


 * 10, 33008304, 38490775 are the first 3 to give 6 out of 7. I don't think 10 is unique. I would guess there exist arbitrarily long sequences of primes from n=1. Maybe this follows from Schinzel's hypothesis H but I haven't checked whether the conditions of the hypothesis are satisfied. I only made a poor slow search program. 6 of 7 and 8 of 15 was the best it found. PrimeHunter (talk) 21:46, 20 October 2009 (UTC)

Well, the next thing I'm going to do is tackle Schinzel's hypothesis H in relationship to these questions. Right now, I think you're probably right. Still, we should at least see if we can get the sixth if not also the seventh term of {2,2,2,2,460724,...}, and if lacking the seventh then at least a matching situation to that for 10 (7 out of 8 or 9). For the latter, our current situation is that 10 is alone for 7 out of 12, I remind you (assuming something isn't missing from the data you've given me), so we're a long way off. Pardon me if I'm wrong in thinking I can presume your continued interest. By the way, don't you find it peculiar that the second and third values giving 6 out of 7 are so relatively close compared to how 10 presents itself? Also, it should be much easier to get a better lock on the 9-out-of-the-first-so-many case, with the jump from n=13 to 26 for x=10. Well, off to H. Thanks again.Julzes (talk) 22:37, 20 October 2009 (UTC)

I'm looking to improve the article referenced with more expert opinion. See my comment/request at Talk:Schinzel's hypothesis H. It's trivial that these polynomials satisfy Bunyakovsky's property (consider x equal the prime in question), so we're dealing with an equivalence between the uniqueness of 10 and a refutation of hypothesis H. It won't prove anything, but right now I'm running a batch factorization of the product of the first seven polynomials through the range 2-2311.Julzes (talk) 00:58, 21 October 2009 (UTC)


 * 610357585 is the first to give primes for n = 1 to 6. It's also prime for 10 and 76. I don't know how much more time I will spend on this. PrimeHunter (talk) 01:13, 21 October 2009 (UTC)

LOL. Of course this is just to satisfy the curiosity, so you're free to move along whenever you like. Very good this 610357585. 10 is still in good company up to n=7 and holding its own on the 7-out-of-9 question, but I do suppose that out there somewhere it will get beat, even if nobody ever finds out where! —Preceding unsigned comment added by Julzes (talk • contribs) 01:25, 21 October 2009 (UTC)


 * The search to 610357585 was only for numbers giving 6 primes from 1 to 6. This can be done faster because prp tests are only needed when none of the 6 numbers have a small factor. But I'm still using a slow program compared to my usual tuple sieve for linear expressions. And there is another unrelated record I want to use cpu time on retaking with a better program. PrimeHunter (talk) 01:52, 21 October 2009 (UTC)

Oh, I see you are really into it there. Good luck with that then (when you do quit this). When this batch process finishes (I think it will be reasonable time-wise, though I wish I had a better approach with programming competence of my own), maybe the data I get could make a simple quick program for up to n=7 easier for one of us. I'm planning to get a good lock on the constant C in the conjecture whose name you can recall better than I. I would then like to invert the logarithmic-power integral for good estimates. If you don't happen to know a quick way or source that you can share for that, I may post to the reference desk for someone else to help. I want to be sure to know when to quit searching before I waste a lot of time.Julzes (talk) 02:23, 21 October 2009 (UTC)

Mr. Sloane has informed me that my recommendation to include the sequences {4,10,73,100,...} and {4,10,1000,...} has been accepted. They are A152396 and A152397.Julzes (talk) 04:57, 21 October 2009 (UTC)

The conjecture I couldn't think of is of course called the Bateman-Horn conjecture. It looks as though the next n with x=33008304 (after n=7) is 362, but that awaits the slow workings of Mr. Alpern's applet.Julzes (talk) 05:21, 21 October 2009 (UTC)

Well, now after suffering my first major crash two days ago my computer automatically shut itself off at the prompting of Microsoft for one of its updates. I have to start the batch process I mentioned all over again, and I lost the answer to whether the next value of n for x=33008304 is 362. I am a little sick right now over this, but I'll be all right.Julzes (talk) 15:50, 21 October 2009 (UTC)


 * I have also experienced unwanted restart by Windows after installation of an update while my pc was searching primes. Later I disabled automatic installation of Windows updates. PrimeHunter (talk) 02:24, 22 October 2009 (UTC)

I just thought I'd drop by and let you know that I verified that 1000 is really the third term after realizing my reasoning had been somewhat (really very far) off. I haven't found the fourth term yet. Also, it looks like I may be getting a four-digit base-seven analogue of 82 in base ten (where this conversation started), but I'll have to wait it out a few more days to be sure. Someone with real computer mathematics skills who's been largely devoted to the 3n+1 problem may come here to ask what exactly the software is for primality testing. I joined a yahoo group on the subject, but I really barely know my way around programming as of this date and am still reliant on Mr. Alpern's applet.Julzes (talk) 22:50, 30 October 2009 (UTC)

I produced an interesting result at the help desk. After about five full days of using the applet I'm accustomed to, I have determined that there is as a final result a 16002-digit binary number that grew from right to left beginning at 1 with the rule that each expansion should be the smallest that is relatively prime to all of its predecessors. I ceased checking all sizable prime factors after a certain point; but, having checked through 5-digit base-ten primes and found that only ones of 4-digits or less ever had an impact, I'm pretty certain I didn't miss anything. A computer program utilizing the Euclidean algorithm will eventually confirm the result, as the most efficient and fully certain means. I have placed the full result in compressed form at the help desk, not having any better means of exposing the result.

What happens initially is that 3, 5, 17 and 257 each double the lengths of the shortest strings of 0s; and then in the end the primes 7, 13 and 97 conspire to each cut away a third of all candidates to continue (as factors of 2^48-1).

I haven't decided what exactly to do with the result--more general questions of the sort?--but it was interesting while it lasted. Absolutely astonishing is the appearance of seven 1s in a row not once but twice, and very far into the sequence the second time.Julzes (talk) 07:31, 3 December 2009 (UTC)


 * It took me a few minutes to make the below PARI/GP script which computed the sequence and printed the number of bits up to 16002 in 20 GHz minutes. It used brute force with no theory to skip numbers. It did not prove whether the sequence had terminated but just stopped when it found no continuation in 1000 steps. 10^6 was an arbitrary limit which was chosen to be large enough. While loops with no limit would have been prettier but this was quick and dirty programming. If you are going to investigate computational number theory then I recommend learning a more flexible tool than an applet designed for one purpose. I often use PARI/GP for easy flexible math programming but usually not when speed is critical. PrimeHunter (talk) 04:56, 5 December 2009 (UTC)

? { v=[1];b=1; for(i=1,10^6, print1(b", ");  for(e=b,10^6,n=2^e+v[i]; if (e==10000,print("\n"i" terms found, ending with "b" bits.");break(2)); for(j=1,i,if(gcd(n,v[j])!=1,next(2))); v=concat(v,n); b=e+1; break; ); ); }

1, 2, 3, 5, 6, 9, 10, 11, 14, 15, 24, 25, 26, 31, 32, 33, 34, 39, 40, 41, 42, 47, 48, 49, 50, 55, 56, 61, 66, 71, 72, 73, 90, 91, 92, 93, 94, 95, 96, 121, 130, 147, 164, 165, 166, 183, 248, 265, 298, 299, 348, 397, 446, 447, 496, 545, 546, 579, 596, 613, 630, 647, 648, 649, 666, 667, 668, 701, 750, 783, 816, 881, 882, 915, 916, 949, 950, 999, 1000, 1017, 1034, 1051, 1100, 1133, 1198, 1199, 1200, 1 217, 1234, 1267, 1348, 1365, 1430, 1447, 1464, 1465, 1466, 1467, 1468, 1501, 153 4, 1535, 1568, 1649, 1666, 1667, 1732, 1733, 1798, 1879, 1880, 1913, 1930, 1931, 2012, 2093, 2110, 2127, 2128, 2145, 2162, 2163, 2196, 2197, 2278, 2295, 2296, 2 297, 2330, 2331, 2428, 2477, 2526, 2527, 2528, 2561, 2594, 2611, 2628, 2629, 264 6, 2663, 2664, 2697, 2746, 2747, 2764, 2765, 2782, 2815, 2816, 2817, 2850, 2915, 2964, 2997, 3014, 3015, 3016, 3017, 3034, 3051, 3084, 3117, 3166, 3183, 3264, 3 313, 3330, 3347, 3348, 3365, 3398, 3447, 3448, 3481, 3498, 3499, 3500, 3517, 351 8, 3519, 3520, 3633, 3682, 3715, 3732, 3733, 3734, 3751, 3752, 3785, 3802, 3803, 3900, 3917, 3934, 3951, 3952, 3953, 3954, 3955, 3972, 3973, 3990, 4055, 4056, 4 137, 4170, 4251, 4252, 4285, 4318, 4335, 4336, 4353, 4354, 4435, 4612, 4645, 472 6, 4839, 4840, 4857, 4938, 4939, 4940, 4941, 4974, 5023, 5056, 5169, 5170, 5171, 5172, 5173, 5174, 5175, 5192, 5369, 5434, 5499, 5500, 5533, 5646, 5679, 5712, 5 729, 5730, 5747, 5844, 5861, 5878, 5943, 5976, 6105, 6122, 6139, 6300, 6301, 630 2, 6303, 6320, 6337, 6402, 6467, 6548, 6549, 6598, 6599, 6632, 6665, 6698, 6715, 6716, 6717, 6718, 6719, 6800, 6833, 6850, 6851, 6900, 6997, 6998, 7063, 7128, 7 209, 7274, 7291, 7356, 7357, 7390, 7615, 7680, 7681, 7698, 7699, 7716, 7781, 781 4, 7847, 7896, 7913, 7930, 7995, 7996, 7997, 8238, 8255, 8272, 8289, 8290, 8291, 8308, 8309, 8390, 8407, 8424, 8425, 8506, 8539, 8604, 8637, 8638, 8687, 8688, 8 737, 8738, 8755, 8756, 8773, 8806, 8807, 8808, 8889, 8906, 8939, 8972, 8989, 900 6, 9039, 9136, 9153, 9170, 9203, 9204, 9269, 9350, 9351, 9352, 9369, 9402, 9403, 9404, 9405, 9486, 9487, 9520, 9521, 9570, 9571, 9572, 9653, 9670, 9671, 9704, 9 737, 9802, 9803, 9804, 9821, 9838, 9855, 9888, 9969, 10002, 10115, 10148, 10181, 10198, 10199, 10200, 10201, 10202, 10251, 10252, 10269, 10270, 10287, 10288, 10 305, 10338, 10435, 10500, 10501, 10518, 10583, 10712, 10729, 10746, 10811, 10876 , 10909, 10990, 10991, 11040, 11121, 11138, 11139, 11252, 11317, 11318, 11367, 1 1384, 11385, 11386, 11387, 11404, 11405, 11406, 11407, 11616, 11617, 11618, 1171 5, 11716, 11717, 11750, 11767, 11768, 11785, 11850, 11867, 11900, 11981, 12046, 12063, 12080, 12129, 12178, 12243, 12244, 12293, 12326, 12327, 12376, 12393, 124 42, 12475, 12540, 12541, 12686, 12703, 12800, 12817, 12818, 12835, 12868, 12885, 12886, 12983, 13016, 13033, 13114, 13339, 13356, 13389, 13422, 13487, 13504, 13 585, 13586, 13587, 13620, 13621, 13670, 13719, 13784, 13865, 13866, 13883, 13900 , 13965, 13966, 13983, 14032, 14065, 14098, 14131, 14148, 14165, 14182, 14215, 1 4216, 14233, 14266, 14315, 14332, 14333, 14350, 14367, 14384, 14385, 14386, 1440 3, 14452, 14453, 14454, 14471, 14472, 14473, 14522, 14539, 14540, 14589, 14638, 14639, 14736, 14817, 14834, 14867, 14868, 14869, 14918, 14919, 14968, 14969, 150 02, 15051, 15180, 15181, 15214, 15263, 15296, 15313, 15378, 15411, 15412, 15413, 15446, 15495, 15576, 15593, 15594, 15611, 15644, 15661, 15678, 15775, 15984, 15 985, 16002, 553 terms found, ending with 16002 bits. time = 7mn, 2,453 ms.


 * I see you have a posted a variation with bits added to the end at the reference desk. Below is the corresponding PARI/GP script and output, ending with the decimal expansion of the last computed number. I have not examined the prime factors or proved whether the sequence really ends there. PrimeHunter (talk) 06:30, 5 December 2009 (UTC)

? { v=[1];b=1; for(i=1,10^6, print1(b", ");  for(e=1,10^6,n=v[i]*2^e+1; if (e==10000,print("\n"i" terms found, ending with "b" bits.");break(2)); for(j=1,i,if(gcd(n,v[j])!=1,next(2))); v=concat(v,n); b=b+e; break; ); ); } 1, 2, 3, 5, 6, 9, 10, 11, 12, 13, 18, 21, 24, 31, 34, 37, 40, 43, 50, 57, 64, 71, 78, 85, 92, 103, 106, 121, 128, 135, 146, 153, 160, 167, 174, 205, 220, 227, 2 50, 265, 280, 303, 318, 333, 356, 379, 402, 425, 448, 487, 494, 501, 508, 515, 5 22, 529, 536, 551, 566, 581, 588, 595, 602, 609, 624, 639, 646, 653, 676, 691, 6 98, 721, 736, 751, 790, 797, 804, 811, 834, 841, 848, 879, 910, 925, 940, 955, 9 86, 993, 1000, 1031, 1038, 1053, 1068, 1075, 1090, 1105, 1120, 1135, 1158, 1173, 1180, 1195, 1226, 1233, 1240, 1247, 1254, 1269, 1284, 1307, 1322, 1337, 1376, 1 399, 1422, 1461, 1484, 1499, 1506, 1529, 1536, 1575, 1622, 1629, 1636, 1651, 167 4, 1697, 1712, 1719, 1734, 1757, 1764, 1787, 1922, 1937, 1968, 1975, 1982, 1989, 1996, 2051, 2122, 2137, 2144, 2151, 2174, 2197, 2220, 2235, 2250, 2265, 2272, 2 319, 2342, 2349, 2396, 2411, 2418, 2433, 2448, 2519, 2526, 2549, 2556, 2571, 259 4, 2601, 2608, 2623, 2694, 2701, 2716, 2731, 2746, 2833, 2840, 2863, 2870, 2893, 2908, 2923, 2930, 2953, 2968, 2991, 3070, 3077, 3084, 3107, 3122, 3233, 3256, 3 263, 3278, 3293, 3300, 3315, 3362, 3393, 3424, 3447, 3470, 3493, 3500, 3507, 365 0, 3673, 3712, 3719, 3726, 3765, 3796, 3867, 3874, 3881, 3888, 3903, 3910, 3917, 3924, 3955, 3962, 3977, 3992, 4007, 4046, 4069, 4076, 4083, 4098, 4121, 4128, 4 135, 4334, 4373, 4380, 4395, 4498, 4505, 4512, 4519, 4526, 4549, 4580, 4587, 459 4, 4609, 4632, 4639, 4646, 4653, 4772, 4787, 4802, 4833, 4856, 4879, 4894, 4909, 4916, 4947, 4954, 4985, 5000, 5015, 5022, 5029, 5052, 5067, 5106, 5113, 5136, 5 151, 5214, 5221, 5228, 5259, 5330, 5337, 5344, 5359, 5382, 5389, 5404, 5419, 543 4, 5449, 5456, 5463, 5486, 5493, 5524, 5539, 5546, 5553, 5584, 5591, 5614, 5621, 5636, 5707, 5714, 5721, 5744, 5759, 5822, 5829, 5836, 5843, 5850, 5873, 5888, 5 919, 5942, 5957, 5972, 5979, 5986, 6001, 6024, 6047, 6054, 6061, 6068, 6107, 614 6, 6169, 6192, 6207, 6222, 6253, 6276, 6291, 6298, 6305, 6336, 6343, 6350, 6357, 6396, 6403, 6434, 6441, 6448, 6463, 6478, 6525, 6548, 6563, 6594, 6601, 6608, 6 623, 6654, 6661, 6692, 6755, 6786, 6817, 6888, 6895, 6926, 6957, 6972, 6979, 699 4, 7009, 7016, 7047, 7070, 7077, 7084, 7091, 7098, 7121, 7200, 7207, 7230, 7293, 7308, 7323, 7450, 7529, 7560, 7591, 7638, 7669, 7780, 7843, 7858, 7873, 7920, 7 967, 8062, 8157, 8172, 8443, 8458, 8537, 8616, 8663, 8758, 8837, 8868, 8931, 896 2, 8977, 8992, 9023, 9118, 9181, 9196, 9291, 9386, 9497, 9512, 9543, 9558, 9589, 9620, 9731, 9794, 9857, 9888, 9903, 9982, 10045, 10156, 10187, 10218, 10361, 10 376, 10407, 10422, 10485, 10564, 10611, 10626, 10641, 10688, 10703, 10766, 10909 , 10924, 10939, 11034, 11049, 11080, 11111, 11142, 11269, 11316, 11379, 11394, 1 1409, 11680, 11727, 11790, 11805, 11836, 11867, 11994, 12073, 12104, 12135, 1216 6, 12197, 12420, 12659, 12674, 12833, 12912, 13007, 13070, 13085, 13164, 13195, 13226, 13257, 13288, 13335, 13430, 13541, 13620, 13635, 13650, 13713, 13728, 137 59, 13790, 13885, 13900, 13931, 13962, 14089, 14104, 14135, 14182, 14213, 14260, 14291, 14306, 14353, 14368, 14383, 14526, 14589, 14604, 14827, 14906, 15033, 15 080, 15095, 15110, 15189, 15316, 15363, 15410, 15553, 15600, 15647, 15758, 15789 , 15820, 15899, 15946, 15961, 15992, 16023, 16054, 16117, 16132, 16227, 16290, 1 6417, 16480, 16543, 16574, 16605, 16684, 16763, 16794, 16937, 17000, 17111, 1714 2, 17173, 17348, 17379, 17410, 17537, 17584, 17599, 17678, 17725, 17820, 17867, 17882, 17945, 17976, 18023, 18054, 18085, 18100, 18131, 18162, 18257, 18272, 183 35, 18558, 18781, 18812, 18955, 19018, 19081, 19160, 19191, 19222, 19269, 19300, 19331, 19362, 19393, 19440, 19455, 19518, 19549, 19564, 19627, 19658, 19769, 19 784, 19799, 19814, 19893, 19988, 20003, 20082, 20113, 20176, 20271, 20302, 20397 , 20412, 20619, 20714, 20761, 20808, 20935, 21206, 21285, 21364, 21491, 21634, 2 1681, 21696, 21727, 21742, 21757, 21772, 21787, 21818, 21865, 21896, 21911, 2192 6, 21941, 21956, 22195, 22258, 22273, 22304, 22383, 22414, 22445, 22588, 22619, 22634, 22697, 22728, 22743, 22790, 22885, 22948, 23011, 23090, 23185, 23200, 233 27, 23374, 23469, 23500, 23691, 23706, 23721, 24040, 24071, 24102, 24133, 24212, 24339, 662 terms found, ending with 24339 bits. ? ## ***   last result computed in 10mn, 44,937 ms. ? print(v[662]) 54389819765939322287141102051500093531930604811702423957158143165137766916180559 40538791956824789909324525229305392830713004389741858256583481279414914182121681 73502979386719967415870921523147834155502524502620960492477104014054651218305831 04129822801944665285262498834242107575680115541480346800208624835586148789604507 08800598441168349238304718219731147846093620375776814998408430848850450041659349 07148878694962135872391250136069131977583614424123870978679098538629930163720029 40630575898459958176351375872885437630750364914464389726011439774893630796287639 75608861186728398883340740161119225982355383934346117271559389608578527216721606 16349924553644908725149334758399222481234270096860146099475723153407805154668458 75410180898772952644776706688592941732769488401940466768092498695166702121503161 40194514924627106551914860432622817907198072354604403060494293333190248287824866 59575849001136261372995271740464141815003309043847691552998743036073414450370983 95226831906912485831945433374084899837203836247183792678900170204394491717427628 05028676801938151530492651715622414662677538949041799827292019769816741134345970 74192114331551837568156501522901494803793463981847993394148673223892927407123223 49003173973541323107015812265892922496353525939477134392908713993233330805027988 34923774587287667512636850348205445535544104932371069632315892513157819546225463 27861292521684682044468888418369133458996992960266878203857246560299514865100567 50124183577581756188858206942864334386498310324561233637798601254029723014218007 17005165150676786536563673136267103211324531979335774371995630411145751144145951 44808068753931923900717862075884726070190367937758304587376973110670908108546888 48216833889393127718693651133675388194096235025750142334670915649233856966901319 18284476552274227358733652956119027382201499880576796174415646099006228302854014 97235229406185407274097215936603591150567095788403751183080686698412800411750911 66075223518152746212284016449879996940040535306310926128305863848607425677843092 92430317236235909626238699920607977843673527674660938984118211440527259373175122 19746032290618611561640985629514200901848302932304668940672809907050000954740223 67302870604888955255633560376937482382025826454713999579981228924273092000954239 50655716844659379381994407297096108117972932818879461111155494233490389194924084 97067007352506976469015828407950868052318983288012855647184326937058921624938661 48591011941605018440014479655083279322602151542765975510576328253917796352594804 95236633580722464900515219360869539274119756600402909125420586495793784999391045 66872149853223482318621727207558910523016239584447974444031140791137818964255253 30571282291474717412766872654598347363439290736211840138910621535683242510060266 58391276627866576662400013829154257847631958480859976684949876919054845941777542 26397562447478368083378996803817758972544755670701347109203727696880887251366214 23705983809494935981772923198933104727082326520841823205999952419476834375567812 71248014643092594899625204444611768325967267133110719162746088289147220273683931 48613565684470764608838252716116726057094230434218479789240000007702499055387459 68692443080628563591556859083183838095486091040027743684880804933061259519401946 51746645881338009862807421413493028837171995734432736843533168595009699875536204 79995417546169660210996354211030464620149112909016603096266210632609293402775568 94360038180098735205241688964693947106706701688122255875439533525836539313195310 50284185631726583121644405741483543581067330963404685825683277841278341837344362 71960350735072978267970194386089341514299179263651619063228710990658932339376825 86922026160735528146576883826902841695150370849637337038622440603404322227617818 00949458043435606102322915685060420939619423974903652006806326138309826684603938 43262019216715961386853712753278292969539799881924488207751097053524167960080029 82030522216649611151717859233400310826535827298519590632025668848028236909484406 13951117112154637726138746321070675850932118313579063648489237268718649634101066 63844755601112994111352389473356663153409045140544920587859701025616264714303748 25175801481760999506046712968159041090314698681671973559557543550877560139835387 05266518433491899827577504388391736868098799018207952590942231112995647823470797 94941831613724113695495187339233843753372146100131842288864188945390744991996797 34747078646523977878815904297337910872615163453810566284910152737139807293465646 91268735456180369183647248852505198338540923106243078540681103309554886459101473 82392489696935274366208590967335490806057629171911874069870358817076182352754380 46674127689578971390885424694984876680153241976858920411191887670748359542935476 15714365826951194740747556245491756273969625261680784012032241373394544099452534 12081980183518749778519669243261521925135369225040900825338704508477157825597838 56529935197589138720918896451606276857624315778585436254632784972983243097711326 32504130978387311256756846342181559171003260966110984325456590659364979874010324 77960323973460520311436137434989050445983999657011051242968674689009178078597855 06116878653700172269926065870067582789151869724193113891717972877800619552153112 66497827357544333802104932276791920495813001563608456869019941197599679240529038 59035157728977331292638536303704203624961127329764229738600056687575984752044768 91493164829907051807650435666158891957158105980624342808935730376357380216014922 24691780916662516932599478247608968318048953416722669648049138550515579015795775 75414700094254239554893587761943798305496629677329610206473373282303442152917600 62308903885466832099539755494111024579471419399404167113150505175972844326931278 58578263978621003238816805594044619474237003694880021351032384064095997726650464 54099962328875536689563832575263430264326667820576049696543230944292772241646109 69909302412637003298019413196363947365039326163728605644876180680498391361441967 93930257882826429536843837245659508294325939194785919947427336954549403626571443 63796805635414503228602929542023942195406140914061920222269309747211537694234514 10850882430070465717904851408415941591551313233974128381658702410887577574966999 33299655174224139781457103822349914276408518113833512303760458876487443202681178 71987828318324432308722054546512504552630507403355467405323876292020698681546467 04747347411898227208212158513976708586401614353171224955819376961912351365215203 52977212231792612404709369639366883668196493799092056569802947000074513342326368 49772347556111848526555961716174503460492953382827950387488388319254770713246537 89355834054133312673314869430298626902957932349915812833084792328698491341329548 66382375860247981474847002151021380815388864182615358577386355851945954228070121 21320182701966480331021436243929146306189925760760174806591784699690134233703751 14199015543004685540920511096309949706085966439389417450940541803128647482681470 21530805890790687043122898210844428576082746240855901879600668591490262103503191 00324774970676043168068829229858107844015529684867905692596858487802388242585342 92063817135368684588218233668388757541555892945434997033998046092063764420341877 38338755849161891224906022366207659092300636111486226725015896877034495935890311 23804561979771424845342934813215308667830652628717243538576400510971189003302846 53129180247396435413903568480796604071808439693214295030081083122654106368631041 02861627986216060867359638730815884470251421697 ?

You're definitely right about learning the programming. Thanks for the confirmation of my first and for preventing my repeating the effort with the second. The way I would have preferred for the first was proof using the Euclidean Algorithm (My notes were as good as certain programming for what you did, though much slower). I'll take care of that when I get around to it. I don't really expect to do a whole lot of computational number theory, but enough to make learning to program for it worthwhile. I also essentially solved the question of the finitude and calculability of the base-3 analogues, with the answer that there is a 50% chance that there are more than 39.3 billion 2s in either one of them (seperately). Back to what I was doing before, I quit trying to extend the sequence {4, 10, 1000, ...} using that applet at 10 million.Julzes (talk) 13:22, 5 December 2009 (UTC)


 * PARI/GP has hundreds of built in functions. My above scripts simply call PARI/GP's gcd to test whether numbers are relatively prime. I don't know whether PARI/GP uses the Euclidean algorithm in gcd. PrimeHunter (talk) 13:47, 5 December 2009 (UTC)

Oh, I see you actually did do a proof (I think the good old Euclidean algorithm is as efficient as anything, so it probably uses it). I misinterpreted your limit of 10^6. The way I did it for a good fraction of the time was to just check for common factors up to 10^6, and that's what I thought you repeated. That's pretty quick work. I wonder if the base-three case might actually be feasible with today's technology. There's not a whole lot of point to it, and straight Euclidean algorithm on all pairs seems unlikely to work (to say the least); but perhaps a specialized optimally designed program for the specific problem running on a fair-sized network could actually handle it in, say, a year. Kind of silly to contemplate. Oh, well, I'm off to consider the theory of higher bases.Julzes (talk) 21:53, 5 December 2009 (UTC)

Allow me to clarify what's been done here. A proof for the first case can be drawn from your program results combined with my analysis of the endpoint factors. The second case is certainly likely to be correct, but is not proven.Julzes (talk) 03:11, 6 December 2009 (UTC)

I have tried to work out my most recent problem at the help desk a first time just using that applet, but kept making errors and quit. I had intended to check whatever I did with PARI/GP anyway, so now I have downloaded that. I'm hopeful that I can find and make use of the manual. I see that PARI/GP is a part of something called SAGE that should be the best thing of its type around because of its open-source nature. I look forward to getting that eventually also, but since it requires a LINUX emulator I'm holding off on that. If I have any problems with the PARI/GP, I may ask your advice.Julzes (talk) 19:42, 7 December 2009 (UTC)

I'm trying to simply duplicate the result from your (unhidden) program above and running into trouble because of variable declaration. Simply typing in what you gave gives a "v=[1]: unknown identifier" response. I assume you declared variables outside the question. I tried to just guess how that might be done, but that didn't work. If I could have a clarification, it might be enough for me to write the little program for my base-4 question. It might not, too, since I want a 2x120 array with the first dimension giving the pair (exponent, coefficient)--I assume that no more than 120 terms are possible from my experimentation. I hate to trouble you with this, but it would be a whole lot quicker than finding the right place in the documentation if you could get me jump-started. I also can't understand the "if(e==10000..." line. You must have had 100000 there, and it seems to belong outside the for loop as just its natural conclusion anyway (I assume this latter fact has to do with  it being a rush job, but maybe I'm misunderstanding something).  Well, anyway, thanks for your trouble in advance, if you can head me in the right direction.  I don't expect an immediate response.  On your time, even if it's the first thing you do tomorrow I might have already made the necessary progress.  I'm not sure, though.Julzes (talk) 21:49, 7 December 2009 (UTC)

I think I figured out the "if(e==b..." line. It seems like you must have had e-b there to get the result.  It makes a little more sense that way, and the arbitrary 10^6 of the for loop will have put a definite stopping point on the thing, as needed.  You then must have simply duplicated the first program for the second where you might instead have made the for loop have 10000 as its stopping point.  I think I understand what you did now aside from the variable-declaration problem I'm really stuck on right now.Julzes (talk) 22:08, 7 December 2009 (UTC)


 * I copied the above from an interactive PARI/GP window. The initial '?' character is a PARI/GP "prompt" which should not be included when you copy the script into PARI/GP. Including it gives "v=[1]: unknown identifier". Variables don't have to be declared. You are right the above output is not from "if(e==10000". I copied something from the wrong place. I actually had "if(e==b+1000" when the output was produced but "if(e==100000" should also work. And yes, this was a rush job and not pretty use of loops. If you paste the corrected version directly into the window started by gp.exe then you should get the above output. How to paste into PARI/GP may depend on your operating system and gp version but if normal methods don't work then try right clicking on the top bar in the window and look at the menu. PrimeHunter (talk) 04:43, 8 December 2009 (UTC)

Thanks for clarifying. I have already made significant progress, and I should have something I can run to solve the problem soon.Julzes (talk) 09:43, 8 December 2009 (UTC)

I had a power outage and also a false result I wouldn't have recognized as such without having experimented, but I finally got the thing to run right. The solution I wanted is 113 digits ending with 33331333331. I still don't know how to use the editor for that programming, so you can imagine how many times I had to type in almost a full program. Plus we had a power outage, which really upset me since I was setting up to take over Mr. Alpern's record page.

Well, now that I have a decent way to get some of this stuff done, maybe I can get the next entry of {4, 10, 1000,...}.Julzes (talk) 07:25, 9 December 2009 (UTC)


 * When making scripts with more than one line, I use an external text editor and either copy-and-paste it into the PARI/GP window, or save it to a text file and read it in with the PARI/GP command . PrimeHunter (talk) 11:37, 9 December 2009 (UTC)

I'll have to figure that all out for future work, but in the meantime I got the fourth term of {4, 10, 1000, ...}. 2191042021910419, 21910420219104192191041821910417, 2191042021910419219104182191041721910416219104152191041421910413 and 21910420219104192191041821910417219104162191041521910414219104132191041221910411 are all prime. The program took about 10 minutes to run. There's no coincidence with the next number (appending down to 21910407) also being prime, so a fifth term is going to take a while (for someone) to get.Julzes (talk) 16:07, 9 December 2009 (UTC)

Surprisingly,the fifth term is only 1113475000 (2 hours).Julzes (talk) 22:18, 9 December 2009 (UTC)


 * Nice. Problems involving concatenation of numbers often allow the concatenation of 1 number being the number itself. Starting at the prime 10087249723, the concatenation of 1, 5, 7, 11, 13 decreasing numbers is prime. It can quickly be determined there is no such 10-digit prime. Can you figure out why? PrimeHunter (talk) 03:34, 10 December 2009 (UTC)

The concatenation of 11 consecutive 10-digit numbers is congruent to its sum is congruent to 0 modulo 11, and then check the primes just over 10^9 (actually, just 10^9+3, to get no final 5 and to get under 10^9).

Right now, I'm trying to figure out how to generate the sequence of minimal positive polynomial coefficients for relative primality, so that I can submit two more sequences, that one and the smallest positive integers that can be plugged in to get relative primality. Writing up the Euclidean algorithm for polynomials is going to be a little harder than what I've done so far. I have trickier things to figure out than the use of '\' and 'floor' instead of 'int' that delayed my first program a little bit.Julzes (talk) 04:18, 10 December 2009 (UTC)

Taking a break from that more difficult problem, I have what I assume is a rediscovery: 357686312646216567629137 is the largest number all of whose right segments are prime. I'm running the problem in all bases from 3 to 100, but it's taking a very long time for base 18, as it did for base 12.Julzes (talk) 08:43, 12 December 2009 (UTC)

Never mind. It's in OEIS. —Preceding unsigned comment added by Julzes (talk • contribs) 09:20, 12 December 2009 (UTC)


 * You can also search numbers in Google. One of many hits is to my own site which says the prime has been found independently by several people. You are now one of them! http://mathworld.wolfram.com/TruncatablePrime.html mentions a 1977 discovery. Wikipedia has a redirect at 357686312646216567629137 to Truncatable prime. Somebody once made an article for it. I wonder whether this is the largest decimal expansion to once have its own seriously meant article. The largest I can currently find is 2305843009213693951. PrimeHunter (talk) 09:59, 12 December 2009 (UTC)

As long as I don't really know how to focus all my cpu time on the most important questions, I might try to expand the list that's in the link to the OEIS sequence. The number is unknown for just about every base greater than 23, but just base 24 looks like a good month's computations. I'm still waiting on base 18, which will be a few hours, and base 24 is some hundreds of times worse. If you happened to see my new question at the help desk, that computation is stalled on a four-line number (The maximum hasn't grown in quite a long time, but it's still working on the tree somewhere--oh, I see it has recently grown to take another third of a line). I'm going to take another look at some of the stuff we were working on earlier (mostly you). Some of it belongs in OEIS. Numbers that outperform 10 w.r.t. simultaneous sol'n of {x2n+xn-1}, for example.Julzes (talk) 10:50, 12 December 2009 (UTC)

I wasn't thinking clearly before at one point, I just realised. I said the Eucidean algorithm was the most efficient way to get the gcd. That's certainly wrong. It makes the most sense to test small primes first. How large one should go is probably system-dependent, too. I have a question about PARI/GP now that you might be able to give a quick answer to. Is there a simpler, quicker gcd==1 test than actually gcd==1? I mean either the language is on the lookout to implement this simpler test and doesn't actually calculate the full negative case, or there is a special test using some other commands, or one must get dirty with the programming in order to get the efficiency one would like. Any clue? I want that efficiency for my left-truncatable relatively prime and composite numbers question, which actually looks like it's at the margins of solvability.Julzes (talk) 08:32, 13 December 2009 (UTC)

Now I'm not sure. Other than Stein's algorithm for finding gcd, I don't see anything faster than checking that both numbers aren't even first--not necessary for what I'm doing anyway--to get a gcd==1 test. It now seems quite possible--though my gut tells me otherwise--that there is no time-saving for gcd==1 versus full gcd calculation, even for very large random numbers. I'd like not to find a more efficient algorithm in a way, because it would entail some programming difficulty, but I have some sense that checking small primes as common factors would be quicker than Euclid's or Stein's algorithms. Where coprimality of a large number with a collection of already-determined coprimes is in question, it seems very likely that small-factor list maintenance and checking would be preferable as a start to multiple implementations of one or the other standard algorithms. The question of optimal efficiency is not only rather complicated, but also pretty important for the problem at hand.Julzes (talk) 09:31, 13 December 2009 (UTC)

In just base 6, the prime-non-restricted left-truncatable coprimal maximum--if even finite--is growing by leaps and bounds (14 lines of digits so far). Funny, bases 2 through 5 were reasonable: In base 10, the numbers are 7, 18709, 21952296054803582154949067894781101832331189 and 7444858551025390541, with 1, 9, 10608 and 490 terminal nodes.Julzes (talk) 14:37, 13 December 2009 (UTC)

The base-7 result is in. 29735375 terminal nodes, 99 digits in base ten. Apart from that, I have found the most extraordinary simultaneous primality result. Thinking in base 3 to start, II0I1000 is the smallest number in whose base 21, 321, 4321, 54321 and 654321 are all prime, and 7654321 is also prime there. That is, in base ten the number is 367434, and for some reason I wanted to see if there was anything nice about the number in other bases. In base 27, it is (18)(18)(0)(18). I'm still waiting on the next term, like a lot of things.Julzes (talk) 04:11, 14 December 2009 (UTC)

87654321 is also prime for the first time in base 6844073124, and I haven't tried seeing if there's anything else neat about this number.Julzes (talk) 14:40, 15 December 2009 (UTC)

I got something more out of the number 367434 in the same vein, and I'll be writing an article intended for The Journal of Recreational Mathematics (I guess) entitled "The Big Left-Truncatable Simultaneous Primality Multi-Base Coincidence" or something like that. What more I got begins with the discovery that the seven-digit expression with smallest sum of digits which gives left-truncatable simultaneous primality in base 367434 is actually 5132491 rather than 7654321. Two readings in base 10 and using bases 36 and 136 are what's involved. I didn't mention it earlier because it's primarily of interest at h2g2, but 4578, the first base that gets one to 54321, has some nice coincidences. These are that 4578 is 2112 in base 13, (42)(42) (or gg the way I would write it) in base 103, and AI in base 456. 103 is nice as both my score on the 1980 AHSME (in 10th grade) and the first prime never used as a factor in the base-four problem I mentioned recently wherein 3 as a factor was disallowed. I'll be talking to you about some less eerie stuff later, but I thought I should share this with you.Julzes (talk) 23:53, 15 December 2009 (UTC)

Something new and neat: 62818269=3360633=199599111.Julzes (talk) 09:30, 16 December 2009 (UTC)

As far as I can tell, I'm the first person to observe that 1885 is palindromic of all lengths up to six, but I wouldn't be surprised to find that someone beat me to it. 1885=98914=111112=124216=1311314. I'm trying to get something for up to seven digits, but I wouldn't be surprised if I go a month on it and then quit.Julzes (talk) 19:56, 18 December 2009 (UTC)

Sixth term of {4,10,1000,...} is in: I started the search immediately after the prior finding, and I was anticipating a long wait in the 13-digit numbers, having ruled out 12-digits theoretically, but the number has come in as 67483920430, with primes for 2, 4, 8, 10, 14 and 20 concatenated numbers. The next should include the concatenation of 28, I think (I have to check that theoretically). Unless I start using something stronger and faster than PARI/GP, I expect something on the order of a year's wait. On another note, aside from this and the palindrome stuff I have been doing, with many OEIS sequences, I have contributed something unusual in A171810. It is a sequence of polynomial coefficients for the smallest choices to make irreducibles by one-degree increments. I have played around with other starts to the sequence than the constant 1, linear x+1, and quadratic x^2+x+1 without getting anything as interesting as the basic sequence, which has one 4 in it corresponding to the 276th degree and (so far) no more 3s after the 574th. I don't have much of a clue with what is going on with that. I also am a bit perplexed by how much time polynomial irreducibility tests take, and I wonder if there isn't an algorithm that would be faster for small positive coefficients. I may post an inquiry on that to the help desk. Talk to you again later.Julzes (talk) 03:57, 21 December 2009 (UTC)

I have more interesting stuff of an elementary nature. I don't know if you are still reading all this, but if you are and happen to have an answer to whether the following are new, I'd be interested: xn+xn-1+...+C is factorable over Z for n=4, C=12 and for n=8, C=20. I'd be surprised if the first isn't known, and factoring it is easy enough. But I kind of expect the second to be new, and its factorization is no trivial effort.. On another note, things are shut down for an upgrade at OEIS, as I found out when I was going to post a lot of stuff. I'll just have to keep track of it. I was going to place an entry with {7, 73, 1476193, 10087249723,...}. When I do, I will at least say you are responsible for the information on the last. Should I credit you with that or for the actual discovery?Julzes (talk) 02:39, 22 December 2009 (UTC)

I recommend looking up 3360633 at OEIS. It comes up in another context involving palindromes; and where it is the eighth term, 33633 is the sixth. Very, very strange! My own contribution just barely escaped prior discovery also, as you can see from the sequence involving base-9 and base-10 palindromes.Julzes (talk) 08:45, 22 December 2009 (UTC)


 * I'm, still reading but don't have a lot of time to investigate and write. A few notes: An efficient coprime test for large random numbers should test for small common factors before computing a full gcd. PARI/GP has functions working on polynomials, for example:

? polisirreducible(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+20) %1 = 0 ? factor(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+20) %2 = [x^4 - x^3 + x^2 - 3*x + 4 1]

[x^4 + 2*x^3 + 2*x^2 + 4*x + 5 1]
 * I would expect this to be known but don't know where to search for it. I computed 10087249723 by myself. There are no Google hits on the number so I guess it's new. PrimeHunter (talk) 12:59, 22 December 2009 (UTC)

'factor' doesn't show up in the help query ('factorpadic' does), and it didn't work right--gave a strange answer--with variable x, only with variable v. I'm going to probably need an upgrade on the PARI/GP at some point, as I have a 2006 version. I probably have been getting 32-bit results on a 64-bit machine, though they've been impressive to me. Anyway, since you don't have time, 3360633 shows up also (with 33633) as a palindrome which is the sum of all composites up to a certain point. As for the coprime test, I think you are right, and I would hope 'if(gcd(m,n)==1,' would be read as a coprime test, but perhaps it is not handled that way in PARI/GP. It would require a bit of foresight and extra programming labor to get that translated properly.  Thanks for the factors.  I had moved on to something else.  x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+20 being factorable is a new small-number coincidence to me, and it seems that only a slightly unusual query brought it out.  You could be right about its being known, though, since it's not too far from the beaten path.Julzes (talk) 20:36, 22 December 2009 (UTC)

I understand that the General Number Field Sieve is the strongest primality testing method, but assume it is hard to implement and that PrimeForm/GW uses something a little inferior but still much better than what I can get out of PARI/GP. Correct this if I am wrong, and also please tell me whether you have any tips that might make my introduction to PrimeForm/GW a little easier than going it alone. I want to try to get the fifth term of {4,10,73,100,...} and PARI/GP just won't cut it. —Preceding unsigned comment added by Julzes (talk • contribs) 23:33, 23 December 2009 (UTC)

By the way, I found out why 'factor' isn't in the help list for polynomials, and that's because it is general purpose including polynomials. The thing about x versus v that I mentioned was probably just my already having defined x on another line. You can tell I'm just getting acquainted with PARI/GP.Julzes (talk) 00:33, 24 December 2009 (UTC)


 * Download PrimeForm/GW from http://openpfgw.svn.sourceforge.net/. If your candidates have not been tested for small prime factors then use the -f switch for trial factoring. General Number Field Sieve is an integer factorization method and not suited for primality testing. PrimeForm/GW is good for primality testing of large numbers. It can make fast probable prime tests of any form and prove primality of certain forms but not concatenation of integers. If you want to end up with proven primes then start with PrimeForm/GW to get probable primes and then, if the form cannot be proved by PrimeForm/GW, use PARI/GP isprime(x) or Primo at to prove primality (don't use isprime above 1000 digits). PrimeHunter (talk) 01:23, 24 December 2009 (UTC)

Thanks. I misremembered what the GNFS was for. I'll see what I can do with the advice, but I sort of expect to give up short of my goal for a few years. Probably by then the programs will have changed substantially. I think I might be better off just understanding what the programs do and working some on writing programs myself. I've taken some graduate number theory courses in the past, and I see no reason I shouldn't become more theoretically competent rather than just technically proficient at using other people's work, though I don't plan to specialize in number theory. While I'm here, I might as well pass along something mildly interesting that another mathematician told me about. An article appeared on the number 3435 and numbers of its type (called Munchhausen numbers for some reason). I couldn't get the article, but it looks like this number is the only one of its kind (aside from 1) in base ten and, perhaps, is company with only one other (base 20) four-digit number. 3435=3^3+4^4+3^3+5^5. Just another little piece of number trivia for you. Julzes (talk) 09:23, 24 December 2009 (UTC)


 * I get http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.3038v2.pdf which calls them Munchausen numbers with one 'h'. Using the convention that 00 = 0, perfect digit-to-digit invariant also mentions 438579088 = 44+33+88+55+77+99+00+88+88. PrimeHunter (talk) 11:29, 24 December 2009 (UTC)

I see. My problem with pdf's is that I have to download something to read them. I'll try that when I wake up, though not especially for this particular subject. I was going to use the convention 00=1, since that is consistent with the limit of xx. I guess I'll be on original terrain with that (if I do it).Julzes (talk) 12:43, 24 December 2009 (UTC)


 * pdf files are common and I definitely recommend downloading the free Adobe Reader. Defining 0^0 = 1 is more common. The pdf does that. PrimeHunter (talk) 13:17, 24 December 2009 (UTC)

I need tech support for this. It seems like I can install whatever I feel like, but then can't actually use it. I downloaded Adobe Reader, but then I wasn't given it as a choice for reading the file. It's not something you can help me with. I'm just a target for spending money by Microsoft and Adobe, in my opinion. I tried Adobe Acrobat before and couldn't get that to function as anything but an advertisement for their products.Julzes (talk) 23:56, 24 December 2009 (UTC)


 * I have installed the free Adobe Acrobat and Adobe Reader on many computers and never had problems. Did you download and install Adobe Reader 9 from http://www.adobe.com/products/reader/? If your browser doesn't open the pdf file then have you tried saving it to disk and opening it there? PrimeHunter (talk) 00:06, 25 December 2009 (UTC)

Don't worry about it. There is something totally wrong with my computer that I won't try to fix until after several weeks of computations on palidromes and other stuff allows me to shut my computer down. I have saved the pdf. Every installment attempt cancels on account of something else already being in the process of being installed, while all I have is PARI/GP running in several windows. It could just be cpu overuse somehow (it's regularly kicking up over 90%), but I won't have any way of knowing without killing my palindrome and polynomial calculations. There is nothing in the article in question that I will find all that interesting anyway, I'm sure; and nothing else immediately requires my reading pdf's or doing anything else with my computer than what I already am.Julzes (talk) 00:17, 25 December 2009 (UTC)

Incidentally, the first doubly 9-palindromic number is a very long wait, and if I have to give up on it I'll also have to warn any future searcher. Not surprising is that the first 4-fold 6-palindrome is also slow in coming and the first 5-fold 5-palindrome is nowhere in sight. I do know that the first number that is a palindrome of all lengths up to 9 will eventually clear as no greater than 2^56, and I can surmise already from the general lack of palindromes of lengths 7, 8 and 9 in the bases that my program has covered that I am going up to base 127 and the trivial example.Julzes (talk) 00:40, 25 December 2009 (UTC)

Arrgh! I just realized some stupidity on my part. The search just mentioned is foolish. I saw bases through the 50s and low 60s come in slowly, but couldn't see the exponential growth in time. Getting through base 127 would be a doubling in time for each of the nested loops. Ridiculous! Well, I can shut that one down. Unless someone is going to use a more powerful computer (or come up with a better search method than just calculating digits as needed, which is unlikely), there will be no proof that 2^56 is the first palindrome of all lengths through 9.Julzes (talk) 01:00, 25 December 2009 (UTC)

Now I realize I overstated things. I suppose it's only a 32-fold inrease, overall, between the last base in the newer search and the last of the search confirming 2^42; but since the base 63 took a good while (8 hours, I'd guess) by itself, that search isn't worth my while.Julzes (talk) 01:18, 25 December 2009 (UTC)

Generate the record-setters, n, of the minimal value of the ratio of the largest prime factor to the smallest of n2+1 (when it's not prime). It has every n divisible by ten, generally multiple times, after a very short start where that's not the case. Looking for a case that doesn't fit this, doing a heuristic theoretical analysis of how likely it is to be generally true, and proving it, of course, are things I thought you might want to look at the possibility of doing.Julzes (talk) 13:02, 12 January 2010 (UTC)


 * ? print(factor((2*k^2)^2+1))
 * [2*k^2 - 2*k + 1, 1; 2*k^2 + 2*k + 1, 1]

If k is not divisible by 5 then 5 divides (2*k^2)^2+1. Let k = 5*m.

(2*(5*m)^2)^2+1 = (50*m^2)^2+1

50*m^2 is always divisible by 10 and is divisible by 100 for even m.


 * ? print(factor((50*m^2)^2+1))
 * [50*m^2 - 10*m + 1, 1; 50*m^2 + 10*m + 1, 1]

When both factors are prime for large m we get a very low ratio that looks hard to beat for other n^2+1. Schinzel's hypothesis H says there are infinitely many cases where both are prime, but nobody has proved any case of the hypothesis except with a single linear polynomial which is Dirichlet's theorem on arithmetic progressions. I guess there are infinitely many large cases and all of them sets a new record ratio but I cannot prove it.

The largest record not of the mentioned semiprime form may be 88526^2+1 = 1973*1993^2 with 1993/1973 = 1.01013...

If (50*42^2)^2+1 = 88200^2+1 = 87781*88621 = (41*2141)*(13*17*401) had been a semiprime then it would have beaten the above with 88621/87781 = 1.00956... PrimeHunter (talk) 14:38, 12 January 2010 (UTC)


 * Oh, I see, it's a simple matter to explain the roundness (or at least you did that awfully quickly). Right, your conjecture is basically the same as mine.  As noted at the help desk, 32000000 is one of the values of n (You might try to explain that one: 2^11*5^6).  In the starting portion, also, the 4th-6th record-setters are such that the 5th is the product of primes in the 4th and 6th: 516^2+1=449*593, 670^2+1=593*757, 844^2+1=757*941 (in case you didn't notice).  The question of whether there are any record-setters not of the form n=50m^2--I'll have to get an approximate probability on that--it shouldn't be too hard (or easy either).Julzes (talk) 15:39, 12 January 2010 (UTC)


 * Did you see the thing about my conjecturing that 7^43+1 is the only record setter by fixed power that has power greater than 3 and base greater than 4 (It's at the help desk)? It's interesting how frequently 2^2p+1 has a smaller largest prime factor than 2^p+1.  I can't think of a reason that would be so.Julzes (talk) 15:49, 12 January 2010 (UTC)


 * 1575185+1 factors with a very low ratio, and the conjecture is false.Julzes (talk) 13:54, 16 January 2010 (UTC)


 * The conjecture seemed very unlikely. 47298725+1 has a lower ratio. PrimeHunter (talk) 03:55, 18 January 2010 (UTC)


 * Ah, so it does. I'd have to change the conjecture a bit.  Do you happen to know how I can avoid this error I'm getting with PARI/GP?  Factor cannot rename file.  It's making it impossible to get results on this.Julzes (talk) 05:23, 18 January 2010 (UTC)


 * I once got similar error messages from PARI/GP factor but they gave a path to the problem file. I don't remember the path but it was apparently a write protected folder. I have a D: drive and could run PARI/GP from there instead of C:. PrimeHunter (talk) 11:45, 18 January 2010 (UTC)


 * Please let me know if you extended your search for p=5 far beyond n=4729872; because I'm using a trial version of Mathematica on it right now, and there is no point in wasting that on duplicated effort.Julzes (talk) 17:58, 21 January 2010 (UTC)I just got a surprisingly nice record-setter: 18000900.Julzes (talk) 18:25, 21 January 2010 (UTC)


 * That's nice indeed. I made an inexhaustive PARI/GP search to 10^7 with limited factoring effort. It took 7 minutes at 2.4 GHz but risked overlooking record-setters. PrimeHunter (talk) 01:18, 22 January 2010 (UTC)


 * I'm surprised by it, but here is another record setter: 201897606. I'm pretty sure that no lower record setters have been missed (They've been mentioned here).  After starting with ratio 11/3 for 2, all of the lowest ratios occur with five prime factors.  The only possible way I could have missed one of these would be in the unlikely case of a non-squarefree number.  The ratio is getting awfully low.  For exponent 7, it looks quite likely that there is not anything approaching the 43/3 one gets with n=2.  I've got a ratio around 27 for n=3123880, but then nothing lower up to 2^26.  Perhaps, though, it is possible to argue probabilistically that this is just the appearance of things from relatively small n.Julzes (talk) 02:39, 24 January 2010 (UTC)


 * Actually, I should say that I haven't ruled out something with 10 prime factors for the p=5 case, but it seems about as likely as a non-squarefree example. Since I'm on the subject, while looking at alternatives for the p=3 case with more than three prime factors, I found (11*47*83-1)^3+1 gives a product of 9 primes between 10 and 100, and also got ratio under 3 with 3758622960^3+1.  The record setters for p=3 are almost certainly always one of two parametric forms.  It's similar to what you did with the p=2 case.Julzes (talk) 03:07, 24 January 2010 (UTC)


 * I know about Dirichlet's Theorem in intimate detail (or did), by the way. That was the subject of a good fraction of a course I took where we were essentially responsible for lecturing the proof.Julzes (talk) 15:53, 12 January 2010 (UTC)

Just thought I'd say hello and give you a run-down on some new things and things in progress. 1) Prime in the final digits of 9999--999779999159200499899; 2) First occurrence of 6/7 digits the same in n^n (excluding final 0s)--5555575 in very middle of 9696; 3) Ascending Smarandache number from 1 to 173--prime in both bases 3 and 6; 4) Ascending Smarandache primes beginning with 1 in base 95--endings of 2, 49, 58, 74, 97, and 117 (First base with more than 2 with ends less than the base and more than 4 reasonably small); 5) Ascending Smarandache prime in base 316 using numbers through 313 is the only one with a reasonable base other than 12 in base 5 that uses the theoretical maximum number of single-digit parts; 6) KJIHGFEDCBA987654321 is first prime in base 4500 (All shorter such strings are prime in much lower bases (192, at most) and for greater base to be needed you have to go to (110)(109)...321); 7) in addition to adding the sequence containing 10087249723 (your finding), the analogue with ascending numbers prime, {2,983,1327373,12695039657,...}, has been submitted, as has {2,278,1826,4498070,2645182700,...} in analogy with the old {4,10,1000,...}; 8) prp's descending from 373 to 1 and from 1825 to 1 in base 7 and not a hint of ever finding a base-13 case. The latter is one of the things I have running right now. I am also seeking 7/9 primes for numbers other than 10 in the subject that dominated the early part of this discussion; trying to extend your result from several years ago of left-truncatable by pairs of digits (I don't have your result yet after a couple of days); running a similar but more tractable calculation of the largest prime that yields primes when the same length of both ends are removed (This shouldn't take too long); calculating the largest k-almost left-truncatable primes, where a composite occurs k times (This is getting slow quickly: The first two are 319687995918918997653319693967 and 3136248319687995918918997653319693967, and I should be able to get you the third (in case you're interested) in a day or two, but I don't plan to go beyond the 4th). That's pretty much it for now.Julzes (talk) 09:31, 3 February 2010 (UTC)

I'm back with that 3-almost left-truncatable prime: Just prepend 132 to the 2-almost one (I might have noticed the 2-almost one was also 3136248 attached to the 1-almost one and saved typing the 2-almost one out.  Note that these three are unrelated to the one without any composites.  I'll see if the fourth is just another extension in a week or so, I guess.)Julzes (talk) 18:25, 4 February 2010 (UTC)

I have the first number to outperform 10 for the first seven primes of {x2n+xn-1}. 2096681555 generates primes for n=1 to 6 and n=8. I'm going to go ahead and submit the terms for 4 to 7 primes to the OEIS. I'll credit you with the middle terms.Julzes (talk) 08:21, 7 February 2010 (UTC)

Here's something: After 1000999998997996995994993, the next concatenation of eight terms starting with a power of ten that is (probably) a prime begins with 10972. How many digits is that? 7777.Julzes (talk) 23:31, 7 February 2010 (UTC)

I still haven't confirmed this last number is prime, and some of my other projects got suspended by a power outage. Here's something I'm interested in: I rediscovered that 82000 is the first and perhaps only non-trivial number that is written with 0s and 1s in bases 3, 4, and 5. The question is what other non-trivial numbers and triples of bases are there like this. I was also briefly trying to find a number that uses no digit greater than three all the way through base ten, but quit that, assuming probably not. What about no digit greater than 4? This seems likely to have a non-empty solution to me. One thing I did was write a program that determines how many, if any, primes of incrementally descending Smarandache-Wellin type from a power of the base are less than the smallest one of incrementally ascending type (also from a power of the base). This question arose because of the huge value--17--for base ten. The upshot is that my program is stalled on base 67, apparently not able to find a single one of the ascending type.Julzes (talk) 21:11, 13 February 2010 (UTC)

Here's a cool curio I just submitted: 151 is the first number that translates to a prime from base 2 to 3 and from bases 2 and 3 to base 4, and also gives primes for all three going to base 5. There's more on this subject, too.Julzes (talk) 07:19, 17 February 2010 (UTC)

I assume you are reading here only for a while--too busy for more than that. I want to update you a little. My curio at the number 4 is absolutely phenomenal, with some pretty significant material not even included. What's not included, in case you look it up, is that the 4th number on the list (The curio is at '4' because it deals with the 4th and 44th items on a list) converts to base 10 as a prime also twice from base 5, and then it doesn't do so again until base 20 (allowing digits greater than 9 for the translation). At bases 20, 22, and 25, it gives 5, 4, and 2 primes (none for bases 21, 23, and 24).


 * Late correction. At base 20, only 4 primes are generated.  This cuts into the primality part of the coincidence, but only to offset it with more of the coincidence surrounding 4.Julzes (talk) 08:50, 13 April 2010 (UTC)

My curio on 151 wasn't accepted, and I almost threw a fit over that, but I got a related one in on 17 (giving me two on that number). One thing I'm grappling with is whether there actually isn't a maximum number of prime translations less than the triangular number obvious absolute bound when looking at groupings of translations up to base 6 or higher. Up to 2^30, assuming I programmed correctly, there is no number that doesn't give a factor of 2, 3, or 5 in at least one of the translations out of the 10 for the up-to-base-6 case. It would be nice to either have proof that this is always an obstacle, or, more likely possible and neater, a good sieving method to search for the first number that gives 10 primes.


 * Well, that program was off, I'll remark here.Julzes (talk) 07:03, 16 March 2010 (UTC)

Aside from these two things, there is one other more strange thing I tried to get into the curios. I looked at the alphametic for 10987, ALI+BABA+WAS+A=LIBRA and thought to come up with something similar for 100999897. The three 9s in a row told me I couldn't have something too similar, and what I came up with was that the number should represent an anagram of letters spaced as the digits. I submitted one with the mistake WE EXCEED W (not spaced right--4 would be between Z & A) and then found later GO OFF MOON. I tried to submit something with the two combined today, but it was rejected. I can easily understand that though.

Hope to talk to you again sometime.Julzes (talk) 04:13, 9 March 2010 (UTC)

I made another base-10 coincidence discovery as well as rediscovering part of one that is known. The latter is at 98689 at PrimeCurios. As for the former, I've submitted it and expect it to be accepted, but who knows? It's that the first prime that continues to read as prime in bases 11-16 goes on to read as prime also in all bases 17-22 except for base 20. That's 231661. I have really taken up a lot of your space here. Feel free to archive it without my taking offense. I should also get you a barnstar--I will--when I get around to figuring that process out. Julzes (talk) 07:03, 16 March 2010 (UTC)

Next thing I'm going to do is at least get you another barnstar, but first I have one coincidence that's pretty far out. I submitted the following (rejected) curio: "5608951: The first collection of four successive primes giving as many as 9 out of 12 primes by concatenation of pairs is 100 times this prime plus 3 times the 7th-10th primes. The remaining three concatenations are all semiprimes." Beyond that, I've just today found the first case generating 10 primes, and both the other concatenations are also semiprimes. 539423223413, ...31, ...97, and ...509. A really hard problem will be getting 11/12. Perhaps you would be interested in that. I really am lost on how Professor Honaker is choosing. He didn't totally shut me out, but it seems like he may only want the big number stuff from me from now on or something.Julzes (talk) 08:50, 13 April 2010 (UTC)


 * I see http://primes.utm.edu/curios/page.php/539423223413.html has been approved. Regarding http://primes.utm.edu/curios/page.php/17769643.html, maybe you have seen my non-consecutive http://primes.utm.edu/curios/page.php/2327138083.html. PrimeHunter (talk) 05:15, 14 April 2010 (UTC)

Just checking back here for the first time in a while, and I see I missed the previous remark before. I might as well answer: I don't think I caught your contribution. I see the relationship to mine. Been doing very little number theory lately. I'm running a computation of the number of primes comprised of three self-counting digits that I expect to take a few weeks. I have the base-4 to base-9 results, and I have the analogues for two self-counting and four self-counting digits through bases 17 and 7, respectively. The program for five self-counting digits just started running a short time ago, and I don't expect anything more than maybe the base-6 and base-7 results, if that much. That's about the extent of what I'm doing that might have a chance of interesting you.Julzes (talk) 09:04, 9 July 2010 (UTC)

The last reported thing was essentially dropped due to a pulled plug, and I started doing something else number theoretical. I have a bit of incomplete stuff hanging out there, but I've essentially gotten now to the point where I need a supercomputer and the people I know to marry each other on demand. Your full name (spelled correctly) produced a 66-digit prime a day after the woman I thought I was primarily pursuing, a 34-year old, had her single first name--she entrusted me with her correct name, which I believe her to no longer be using as of yesterday--produced a 77-digit prime. There were various checks. I've been in a cycle of increasingly stupendous coincidences in this system I've engaged on. I realise how bizarre this will sound. I hope you get to it first, but probably not. I'm so busy at this point with so many personal and place names that I don't have time to be clear. This note is to let you know personally, since I believe I essentially sent your name in one end of the Pentagon and out the other on this specific topic.Julzes (talk) 03:54, 20 August 2010 (UTC)

Yesterday, I was trying to report a simple indisputable existence proof via expansion of the exclamation DOH with OWE repeatedly substituted for O. When I recover that, I'll get back to you. I haven't taken the best of notes. Too much at one time. This would all be in relationship to some addition to the subject of numerology that I first came here about. I haven't been much of a wikipedian so far. Some day things will settle down and I won't have anything to do but.Julzes (talk) 03:54, 20 August 2010 (UTC)

Left truncatable polynomials
Something else I have been doing appears to have reached the limit of my computer the way I have been doing it, and I thought I'd let you know about it. The sequence {2,3,5,7,8,10,14,15,19,21,24,26, 28,32,37,40,42,46,48,50,57,59,61,67,77,84,91,96 ,...} (I've waited days now for the next term) represents the smallest sum of coefficients of nth degree polynomials that can be left-truncated to give all primes when 2 is plugged in. The way I ended up doing this has produced a program with long strings of additions and subtractions, as well as about sixty close parentheses in a row, because I wasn't competent enough and/or willing to put in the time to write a program that would generate the sequence more intelligently. Despite this, it's gone a lot further than I would have expected. Perhaps you'd be interested in writing a better program for this; one that allows plugging in a value other than 2 and generates the sequence faster by eliminating a lot of unnecessary simple arithmetical calculations, and (harder) by maintaining data from lower degrees in the search for next terms. To clarify what the sequence is, 10 is the 6th term by virtue of the fact that, at least as an example, x6+x5+3x4+2x3+x2+x+1 is prime for x=2 and so is every lower-degree part of the polynomial. I'd have to check, and I am not sure if it ever makes a difference, but I believe I have allowed that the constant term may be any positive integer (It's possible I wrote that it must be 1 or a prime, but I am pretty sure I haven't required that it be 1 (The example for each case has given this constant value so far anyway)).Julzes (talk) 21:17, 14 January 2010 (UTC)


 * I guess all coefficients must be positive integers. What is your 28 solution for degree 13? I only found 29 with these coefficients for x^0 to x^13:
 * 1, 1, 1, 3, 1, 1, 3, 5, 2, 2, 3, 4, 1, 1. PrimeHunter (talk) 13:36, 15 January 2010 (UTC)

It looks like I screwed that one up somehow! I just checked what I had--2,2,1 at the end instead of 4,1,1--and it doesn't work.Julzes (talk) 14:17, 15 January 2010 (UTC)


 * I have also been unable to match your sums for degree 16 to 23 where I stopped. PrimeHunter (talk) 14:24, 15 January 2010 (UTC)

Well, that's what happens when you program it the way I did. I'll have to find the bad code, apparently, when this stops running.Julzes (talk) 14:29, 15 January 2010 (UTC)

I definitely did something with the 11th degree term that I shouldn't have, I see from checking the example for the 16th term of the sequence.Julzes (talk) 14:37, 15 January 2010 (UTC)


 * I get 2, 3, 5, 7, 8, 10, 14, 15, 19, 21, 24, 26, 29, 32, 36, 41, 44, 49, 53, 54, 61, 64, 71. This is 1 lower than you for degree 15 where I have coefficients [1, 1, 1, 2, 3, 1, 1, 7, 1, 1, 3, 2, 2, 6, 1, 3] or [1, 2, 2, 3, 1, 4, 3, 3, 1, 3, 2, 2, 3, 1, 1, 4]. I use a small recursive function with the same code for all degrees. PrimeHunter (talk) 14:47, 15 January 2010 (UTC)

Whatever I did wrong is making less sense to me if you got a lower number at any point. I'll have to see what it was. The same code certainly sounds like the right approach!Julzes (talk) 15:07, 15 January 2010 (UTC)


 * My code:

\\find nth degree when (n-1)th degree has coefficient sum s and value v f(n,s,v)={ local(i,e); if(n>d,print("degree "d" sum "s": "concat(cc,c));if(s<=m,m=s-1;best=s);return); e=x^n; i=1; while(i<=m-s-(d-n),   if(ispseudoprime(i*e+v),c[n]=i;f(n+1,s+i,i*e+v));    i++;  ); }

ff(dd,mm,xx) = { d=dd;m=mm;x=xx; c=vector(d); best=0; j=1; while(j<=m-d,cc=j;f(1,j,j);j++); return(best); }
 * ff(15,40,2) searches for degree 15 with the lowest sum<=40 for x=2. It returns the lowest sum, or 0 if no solution is found. Any positive constant term is allowed. Make a primality test in the last while loop to limit it to primes. PrimeHunter (talk) 15:35, 15 January 2010 (UTC)

I'll have to figure out exactly what your program does, but it's good to be able to use it (I'm still going to have to find out how exactly I messed up on mine). I've already obtained the next three terms with 2 and the first twenty terms with 3.Julzes (talk) 16:43, 15 January 2010 (UTC)


 * ff initializes variables and calls f for each constant term.
 * cc: constant term
 * d: searched degree
 * best: lowest sum found so far, 0 as long as no sum found
 * m: currently searching a sum<=m, m is 1 less than best when a sum has been found
 * c: vector with d coefficients of x^1 to x^d
 * When f(n,s,v) is called, cc and c[1] to c[n-1] gives a valid polynomial of degree n-1 with sum s of coefficients (including the constant cc), and value v at x. c[n] to c[d] are ignored garbage at the time of f(n,s,v,) is called. i runs through potential coefficients of x^n. i<=m-s-(d-n) because the sum s will increase by i if i is picked for x^n, and the next d-n degrees have coefficient at least 1. If i gives a (probable) prime for degree n then set c[n]=i and make a recursive call for degree n+1 with updated values of s and v. n>d at the start of f signals that we already have a complete solution with degree d. n==d+1 could also have been used. PrimeHunter (talk) 17:05, 15 January 2010 (UTC)

You didn't need to go to all the trouble of explaining it; it wasn't going to be that hard. Thanks, though.Julzes (talk) 17:49, 15 January 2010 (UTC)