Wikipedia:Reference desk/Archives/Mathematics/2023 September 27

= September 27 =

Digit sum
If A = 7777777^7777777 remove the leftmost 7 digits and the rightmost 7 digits, the digit sum of A is B, and the digit sum of B is C, what is the digit sum of C? 220.132.230.56 (talk) 04:25, 27 September 2023 (UTC)


 * A is a 53595538-digit number. The leftmost 7 digits of A are 2080865 and the rightmost 7 digits are 5347697. Beyond that I can't help you. --Lambiam 06:59, 28 September 2023 (UTC)
 * Well, 7777777 == 4 mod 9, and 4^n == 4, 7, 1, 4, 7, 1, 4, 7, 1, … mod 9, thus 7777777^7777777 == 4 mod 9, and 2080865 == 2 mod 9, and 5347697 == 5 mod 9, thus A (remove the leftmost 7 digits and the rightmost 7 digits) == 4-2-5 == 6 mod 9 210.243.207.185 (talk) 10:10, 28 September 2023 (UTC)
 * A has 53595538 digits, even if every digit of A is 9, the digit sum is 482359842, thus B <= 482359842, and for the numbers <= 482359842, the number with largest digit sum is 399999999, which has digit sum 75, thus C <= 75, and the digit sum of C must be == 6 mod 9, thus the answer must be 6 unless C = 69 (but very unlikely), in this case the answer is 15 210.243.207.185 (talk) 10:17, 28 September 2023 (UTC)
 * To complete the proof, prove that B < 339999999. --Lambiam 14:27, 28 September 2023 (UTC)

Could you create these OEIS sequences and extend existing OEIS sequences?
There are OEIS sequences:


 * Smallest k such that 2*n^k+1 is prime (A119624, and subsequences A253178 and A098872)
 * Smallest k such that 2*n^k-1 is prime (A119591, and subsequence A098873)
 * Smallest k such that 10*n^k+1 is prime (A088782)
 * Smallest k such that (2*n-1)^k+2 is prime (A138066)
 * Smallest k such that (2*n-1)^k-2 is prime (A255707)
 * Smallest k such that (n-1)*n^k+1 is prime (A305531, and subsequence A087139 with all numbers added by 1)
 * Smallest k such that n^k+(n-1) is prime (A076845)
 * Smallest k such that n^k-(n-1) is prime (A113516)
 * Smallest k such that n^k-(n+1) is prime (A178250)

And could you create these sequences in OEIS? I do not have an OEIS account.


 * Smallest k such that 3*(2*n)^k+1 is prime (with a subsequence in OEIS: A098877)
 * Smallest k such that 3*(2*n)^k-1 is prime (with a subsequence in OEIS: A098876)
 * Smallest k such that (2*n)^k+3 is prime
 * Smallest k such that (2*n)^k-3 is prime
 * Smallest k such that (n-1)*n^k-1 is prime (with a subsequence in OEIS with all numbers added by 1: A122396)
 * Smallest k such that (n+1)*n^k-1 is prime
 * Smallest k such that (n+1)*n^k+1 is prime
 * Smallest k such that n^k+(n+1) is prime

Also, for the existing sequences, A119624 can be extended to n=364 (and thus A253178 can be extended to n=242), A098872 and A098873 can be extended to at least n=171, A250200 can be extended to n=152 (use the A255707 data), A087139 can be extended to n=91, A076845 can be extended to at least n=256 (107^1400+106, 113^20088+112, 123^64370+122, 173^11428+172, 179^3356+178 are primes or PRPs), A178250 can be extended to n>743 (215^22342-216 and 743^14096-744 are PRPs)

Thanks. 220.132.230.56 (talk) 04:49, 27 September 2023 (UTC)


 * For the large terms just see: https://www.rieselprime.de/ziki/Williams_prime_MM_least and https://harvey563.tripod.com/wills.txt for (n-1)*n^k-1, https://www.rieselprime.de/ziki/Williams_prime_MP_least for (n-1)*n^k+1, https://www.rieselprime.de/ziki/Williams_prime_PM_least for (n+1)*n^k-1, https://www.rieselprime.de/ziki/Williams_prime_PP_least for (n+1)*n^k+1, https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n for 2*n^k+1 and 3*(2*n)^k+1, https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n for 2*n^k-1 and 3*(2*n)^k-1 220.132.230.56 (talk) 04:52, 27 September 2023 (UTC)
 * You can request an OEIS account by going to the OEIS website and click "register". Bubba73 You talkin' to me? 02:16, 28 September 2023 (UTC)
 * You said that you have OEIS account, could you add these sequences to OEIS? Thanks. 220.132.230.56 (talk) 15:11, 3 October 2023 (UTC)
 * For one thing, I don't want to do it under my OEIS account because I would be responsible for being sure they are right. Also, earlier I said that you should have at least three terms to start a sequence. The guidelines want four terms (but there are exceptions). See "how many terms do we need" .  Bubba73 You talkin' to me? 22:27, 4 October 2023 (UTC)
 * Well, I have many terms:
 * Smallest k>=1 such that 3*(2*n)^k+1 is prime, for n = 1 to 100:
 * 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 7, 3, 1, 1, 1, 3, 2, 1, 9, 1, 3, 1, 1, 1, 5, 2, 1, 12, 1, 1, 2, 1, 14, 1, 1, 2, 1, 2, 2, 2, 4, 1, 1, 1, 3, 2, 5, 1, 1, 3, 270, 1, 1, 12, 1, 46, 2, 1, 1, 1, 27, 21, 1, 4, 1, 8, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 2, 1, 4, 1, 2, 3, 4, 1, 3, 4, 1, 1, 11, 12, 2, 1, 1, 2, 2, 11, 1
 * Smallest k>=1 such that 3*(2*n)^k-1 is prime, for n = 1 to 100:
 * 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 11, 1, 1, 1, 2, 2523, 1, 1, 2, 1, 2, 2, 1, 1, 1, 59, 1, 1, 10, 2, 2, 2, 1, 1, 1, 14, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 1, 1, 6, 2, 50, 63, 1, 1, 1, 2, 11, 50, 1, 1, 38, 1, 2, 2, 1, 26, 1, 3, 1, 1, 3, 1, 1, 2, 1, 79, 1, 21, 1, 1, 3, 1, 2, 14, 3, 2, 6, 1, 1, 1, 4, 3, 1, 1, 1
 * Smallest k>=1 such that (2*n)^k+3 is prime (0 if no such k exists), for n = 1 to 100:
 * 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 1, 20, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 4, 0, 1, 5, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 5, 4, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 2, 3, 0, 1, 1, 0, 15, 1, 0, 2, 1, 0, 1, 3, 0, 1, 2, 0, 1, 1, 0, 3, 3, 0, 1, 1, 0, 1, 1, 0, 8
 * Smallest k>=1 such that (2*n)^k-3 is prime (0 if no such k exists), for n = 1 to 100:
 * 3, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 10, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 1, 105, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 2, 0, 1, 204, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 2, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 2, 4, 0, 1, 1, 0, 3, 1, 0, 2, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 10, 2, 0, 1, 1, 0, 1
 * Smallest k>=1 such that (n-1)*n^k-1 is prime, for n = 2 to 100: (rifo: https://harvey563.tripod.com/wills.txt)
 * 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 14, 1, 1, 2, 6, 1, 1, 1, 55, 12, 1, 133, 1, 20, 1, 2, 1, 1, 2, 15, 3, 1, 7, 136211, 1, 1, 7, 1, 7, 7, 1, 1, 1, 2, 1, 25, 1, 5, 3, 1, 1, 1, 1, 2, 3, 1, 1, 899, 3, 11, 1, 1, 1, 63, 1, 13, 1, 25, 8, 3, 2, 7, 1, 44, 2, 11, 3, 81, 21495, 1, 2, 1, 1, 3, 25, 1, 519, 77, 476, 1, 1, 2, 1, 4983, 2, 2
 * Smallest k>=1 such that (n+1)*n^k-1 is prime, for n = 2 to 100:
 * 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 1, 4, 3, 1, 1, 1, 2, 7, 1, 2, 1, 2, 1, 2, 1, 1, 2, 4, 2, 1, 2, 2, 1, 1, 2, 1, 8, 3, 1, 1, 1, 2, 1, 2, 1, 5, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 2, 1483, 1, 1, 1, 24, 1, 2, 1, 2, 6, 3, 3, 36, 1, 10, 8, 3, 7, 2, 2, 1, 2, 1, 1, 7, 1704, 1, 3, 9, 4, 1, 1, 2, 1, 2, 24, 25, 1
 * Smallest k>=1 such that (n+1)*n^k+1 is prime (0 if no such k exists), for n = 2 to 100:
 * 1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 1, 0, 1, 1, 0, 1, 9, 0, 1, 1, 0, 2, 1, 0, 2, 1, 0, 5, 2, 0, 5, 1, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 2, 2, 0, 2, 6, 0, 1, 183, 0, 2, 1, 0, 2, 1, 0, 1, 21, 0, 1, 185, 0, 3, 1, 0, 2, 1, 0, 1, 120, 0, 2, 1, 0, 1, 1, 0, 1, 8, 0, 5, 9, 0, 2, 2, 0, 1, 1, 0, 2, 3, 0, 9, 14, 0, 3, 1, 0
 * Smallest k>=1 such that n^k+(n+1) is prime (0 if no such k exists), for n = 2 to 100:
 * 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 2, 1, 0, 1, 3, 0, 1, 6, 0, 4, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 3, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 2, 4, 0, 1, 1, 0, 2, 1, 0, 2, 1, 0, 1, 30, 0, 1, 3, 0, 1, 1, 0, 7, 67, 0, 1, 1, 0, 1, 1, 0
 * 220.132.230.56 (talk) 14:45, 5 October 2023 (UTC)
 * Given your particular interest in sequences, I would highly recommend registering an OEIS account yourself; it makes it easier for you to submit sequences as so desired without having to rely on other people (who may not have the time or incentive to maintain such entries) to do so. GalacticShoe (talk) 14:55, 5 October 2023 (UTC)
 * I agree. If you request an account, it will probably be accepted.  I posted the guidelines for submitting a sequence above.  Other editors will look at your submission, and probably approve it. I'm working on other things, so I have no desire to submit them for you.  Bubba73 You talkin' to me? 19:26, 5 October 2023 (UTC)
 * Besides, new accounts are limited to 3 drafts at a time. So if you ask someone else to submit 3 sequences for you, they won't be able to make any other changes or create new sequences until these are approved or rejected. And, unlike Wikipedia, because of the small number of reviewers, the process can be very slow. Dhrm77 (talk) 21:26, 5 October 2023 (UTC)
 * Probably, but why don't you? —Tamfang (talk) 18:48, 3 October 2023 (UTC)
 * Out of curiosity, what makes any of these sequences interesting? Are they useful for something in the real world? I mean, you can create any sequence being any equation or any intersection of any 2 sequences, ad infinitum, but is any of it useful? Dhrm77 (talk) 15:02, 4 October 2023 (UTC)

Why not define “division” in rings?
e.g. matrix, which have matrix multiplication AB and invertible matrix (multiplicative inverse) A-1, why not define matrix division? A ÷ B is A (B-1)

Also, in the ring Z21, which is not a field, we can define 5 ÷ 2 = 13, since the multiplicative inverse of 2 is 11 and 5 × 11 is 13 220.132.230.56 (talk) 07:03, 27 September 2023 (UTC)
 * You cannot do this in a ring that is not a domain. In a domain, you can do so by constructing the field of fractions, which may still require adding elements to the ring. Z21 has zero divisors in the form of 3, 6, 7, 9, 12, 14, 15, and 18, which means you cannot divide by them. The matrix ring also has zero divisors in the form of non-invertible matrices. "Division" is reserved for situations where we do not have so many exclusions (fields have just one, zero).--Jasper Deng (talk) 07:55, 27 September 2023 (UTC)
 * Even if every nonzero element is invertible (in a division ring; the most famous example is the quaternions) people typically only define division when the ring is commutative. Otherwise, you need to distinguish between "right division" and "left division" (one of them will be $$B^{-1}A$$, the other one $$AB^{-1}$$, and these are typically not the same). —Kusma (talk) 08:13, 27 September 2023 (UTC)
 * So for non-domains such as sedenions and split-complex numbers, there are also no “division” defined? 220.132.230.56 (talk) 15:10, 3 October 2023 (UTC)

Are there n kinds of numerical systems with radix n?
For positive integer n (to be the base (radix) we used), choose n consecutive (positive or negative or 0) integers (must include the integer 1) as the n digits to be used, and there are n ways to choose.


 * n=1: the unary numeral system (1)
 * n=2: the binary numeral system (0,1) and the bijective base 2 (1,2)
 * n=3: the balanced ternary (-1,0,1) and the ternary numeral system (0,1,2) and the bijective base 3 (1,2,3)
 * n=4: we can choose (-2,-1,0,1) and (-1,0,1,2) and (0,1,2,3) and (1,2,3,4) as digits, (0,1,2,3) is the quaternary numeral system and (1,2,3,4) is the bijective base 4

If we choose (-2,-1,0,1) (use A for -2 and B for -1), the numbers 1 to 20 will be written as:


 * 1, 1A, 1B, 10, 11, 1AA, 1AB, 1A0, 1A1, 1BA, 1BB, 1B0, 1B1, 10A, 10B, 100, 101, 11A, 11B, 110

If we choose (-1,0,1,2) (use A for -1), the numbers 1 to 20 will be written as:


 * 1, 2, 1A, 10, 11, 12, 2A, 20, 21, 22, 1AA, 1A0, 1A1, 1A2, 10A, 100, 101, 102, 11A, 110

220.132.230.56 (talk) 07:18, 27 September 2023 (UTC)


 * It depends on your definitions, but to my mind there are an infinite number of number systems with radix 2. Consider the following system: Possible digits are 0, 1, A for -1; consecutive non-zero digits are not allowed. Counting, starting from 1, goes: 1, 10, 10A, 100, 101, 10A0, 100A, 1000, ... . There are variations on this, but the idea is by restricting the possible sequences of digits in various ways you can produce many number systems for a given radix. Note that a radix does not have to be an integer; see Golden ratio base. In that system the possible digits are 0, 1; consecutive 1's are not allowed. I'm not sure that bijective numeration really counts as an arithmetic base since there is no 0 digit. I would expect that you be able to express fractional number in a number system; how would you write 1/2 or 1/3 in this system? In order to shift a decimal point to the right you need to be able to prepend 0 digits, and you can't do that if there is no 0 digit. I have similar issues with the unary number system; it's not the same as base 1, and you can't write fractions with it. --RDBury (talk) 09:16, 27 September 2023 (UTC)

Are there n possible number systems with dimension n, if n is power of 2?
e.g.

n=2: i^2


 * -1: complex number
 * +1: split-complex number

n=4: i^2, j^2, k^2


 * Three -1: quaternion
 * Two -1 and One +1: bicomplex number
 * One -1 and Two +1: split-quaternion
 * Three +1: hyperbolic quaternion

etc. 220.132.230.56 (talk) 07:26, 27 September 2023 (UTC)


 * I'm not sure what you mean by "number system". Does the Classification of Clifford algebras help? —Kusma (talk) 08:16, 27 September 2023 (UTC)
 * I think we can assume "number system" here means an algebra over the reals. But it's important to realize that not every mathematical object has a corresponding article in Wikipedia. I don't know how many there are for a given dimension (there's probably an OEIS sequence for it), but I'm pretty sure that the perceived pattern that number of notable algebras = n when n is a power of two is just a coincidence. Note that the dimension 4 list only includes non-commutative examples, commutative examples include R⊕R⊕R⊕R, C⊕R⊕R, C⊕C. --RDBury (talk) 09:52, 27 September 2023 (UTC)
 * The OP's idea does have substance. These algebras come from the Cayley–Dickson construction and its generalizations, which double the dimension. The standard construction yields the complex numbers, the quaternions, the octonions, the sedenions and so on. Think you would have to look at the refs and original papers to see if some specific generalization yields precisely n algebras for dimension 2^n.John Z (talk) 05:44, 29 September 2023 (UTC)
 * So what are the 8 possible number systems with dimension 8? If all the 7 numbers ek^2 are -1, then this is the octonions, if all the 7 numbers ek^2 are +1, then this is the hyperbolic octonions, but what are the other 6? 220.132.230.56 (talk) 15:14, 3 October 2023 (UTC)

coloring edges of an icosahedron
The 30 edges of a regular icosahedron are randomly colored green and blue. What is the probability that at least one vertex has all blue edges? The obvious answer would be $$1-(\frac{31}{32})^{12} \approx 0.316811438$$ if edges were independent, but they're not: each edge affects two vertices.

The question arose as I was thinking about generating a fractal planet by midpoint displacement. Related questions include: how many ways can every vertex have exactly three blue edges? (I guess that is equivalent to: how many Hamiltonian circuits?) How many ways can each vertex have two or three blue edges? —Tamfang (talk) 17:03, 27 September 2023 (UTC)


 * I wrote a small program to count colorings, and I got $$320002263$$ edge colorations where at least one vertex has all blue edges (out of $$2^{30}$$ possible edge colorations.) Moreover, I counted $$2780$$ colorings where all vertices have exactly three blue edges, and $$3388160$$ colorings where all vertices have exactly either two or three blue edges. I'm not sure if these are accurate though, so I'll wait for someone else to confirm them. GalacticShoe (talk) 18:59, 27 September 2023 (UTC)
 * Note that the probability that one vertex has all blue edges is then $$\frac{320002263}{2^{30}} \approx 0.298025331$$, which is surprisingly not far from the naive edge-independence estimate. GalacticShoe (talk) 20:12, 27 September 2023 (UTC)
 * Thanks. I suppose I ought to have tried that before posting. —Tamfang (talk) 23:07, 27 September 2023 (UTC)
 * Greetings,
 * The problem you've described involves coloring the edges of a regular icosahedron randomly with two colors (green and blue) and then calculating the probability of specific color patterns at the vertices. The key challenge here is that the edges are not independent since each edge affects two vertices.
 * To calculate the probability that at least one vertex has all blue edges, you can use the complementary probability approach. The complementary probability is the probability that none of the vertices has all blue edges. You can calculate it as follows:
 * Let A be the event that a specific vertex has all blue edges, and B be the event that another specific vertex has all blue edges. Since these events are not independent, you can't simply calculate the probability of each event and multiply them.
 * So, to find the probability that none of the vertices has all blue edges, you can consider each vertex individually. For each vertex, the probability that it does not have all blue edges is (1−p)30, where p is the probability that a specific edge is blue (independent of other edges).
 * Therefore, the complementary probability is:
 * P(None of the vertices has all blue edges)=(1−p)30
 * To find the probability that at least one vertex has all blue edges, you can use the complementary probability:
 * P(At least one vertex has all blue edges)=1−P(None of the vertices has all blue edges)
 * Now, you can plug in the value of p (the probability that a specific edge is blue) to calculate the desired probability.
 * As for the related questions about how many ways there are for every vertex to have exactly three blue edges and how many ways each vertex can have two or three blue edges, these are combinatorial questions that can be quite complex. They involve counting Hamiltonian circuits and satisfying certain degree conditions in the graph. The exact answers would depend on the specific constraints and structure of the icosahedron. These types of counting problems often require advanced combinatorial techniques. TheAlienMan2002 (talk) 19:00, 27 September 2023 (UTC)
 * On Stack Exchange, answers generated by chatbots are forbidden, I'm just sayin'. —Tamfang (talk) 23:04, 27 September 2023 (UTC)
 * I was accused of being an AI on a different forum. I'm almost positive I'm not, but you never know. --RDBury (talk) 06:19, 28 September 2023 (UTC)
 * It is more likely that you are a Boltzmann brain. --Lambiam 07:02, 28 September 2023 (UTC)
 * (I guess that is equivalent to: how many Hamiltonian circuits?) No, because there can be shorter loops. —Tamfang (talk) 22:43, 27 September 2023 (UTC)

Limit help
Can someone help me figure this one out, it seems too rigorous.

Consider the function f(x) defined as follows:

$$ f(x) = \frac{3x^2-4x-1}{x-1} $$

Find the limit as x approaches 1 of f(x) TheAlienMan2002 (talk) 21:55, 27 September 2023 (UTC)
 * 3x2−4x−1 approaches −2. x−1 approaches 0. See One-sided limit for how to deal with that when the sign matters. But I would first check the problem really does say –1 and not +1 in the numerator. +1 would give a very different type of problem. PrimeHunter (talk) 22:20, 27 September 2023 (UTC)


 * Are you sure it's not supposed to be +1 in the numerator? That limit would exist. --RDBury (talk) 06:08, 28 September 2023 (UTC)


 * If you have +1 instead as mentioned above then Indeterminate form may be of use. NadVolum (talk) 08:47, 28 September 2023 (UTC)