Wikipedia:Reference desk/Archives/Mathematics/2012 November 7

= November 7 =

bitcoin alternative strategies
i see i will have to do this myself. what alternative to proof-of-work can i use in my bitcoin-alternative scheme? (see above question). --91.120.48.242 (talk) 07:31, 7 November 2012 (UTC)


 * Indeed, I guess no one else is going to ask others to invent something for you. —Tamfang (talk) 08:02, 7 November 2012 (UTC)
 * How about generating a unique previously unknown prime number. These require proof of work and may conceivably be useful.--Salix (talk): 12:21, 7 November 2012 (UTC)


 * Thank you, but can we first concentrate on an implementation-wide fundamental alternative to the requirement for "proof-of-work"? Isn't there some other solution that would conceivable work but simply has not been done?  Do you have any ideas that might work?


 * As for you, Tango, I am actually quite able to produce my own proofs: I am highly familiar with many definitions in mathematics, and I guess you could say the rest just follows logically :) --91.120.48.242 (talk) 13:37, 7 November 2012 (UTC)
 * You have made 0 effort to debate this issue with people who understand it in your previous question on the topic. You are trying to reinvent the square wheel without any understanding of what purpose proof of work serves in Bitcoin. You have ignored my references to proposed alternatives.
 * I'd morally support the development of an alternative currency based on my proof of stake system, but only by someone I trust to do a good job. -- Meni Rosenfeld (talk) 18:49, 8 November 2012 (UTC)

Square Tile Maps and Existence of Adjacent Tiles Equidistant from a given Tile
Given a subset of a rectangular grid of equal sized square tiles can you have two adjacent tiles equidistant from a third? Since I tend to muddle questions when I ask them: when I say subset, I mean that some tiles are removed; a tile is adjacent if it is one tile up, right, left, or down not if it is connected diagonally; distance is measured as the length of the shortest path of adjacent tiles connecting to tiles. I think it follows by a simple application of induction and I can't think of any counter examples, but for some aggravating reason I feel like I might be missing something. Thank you for any and all help:-) Phoenixia1177 (talk) 11:02, 7 November 2012 (UTC)
 * It's impossible. Given a path connecting two tiles, argue by induction on path length that the parity of the path length equals the $$L^1$$-distance between the tiles.--80.109.106.49 (talk) 11:58, 7 November 2012 (UTC)
 * "equals the parity of the $$L^1$$-distance" that should be.--80.109.106.49 (talk) 19:52, 7 November 2012 (UTC)
 * Thank you, makes perfect sense. I don't know why simple problems seem such a pain sometimes. Anyways, thanks again:-)Phoenixia1177 (talk) 11:08, 10 November 2012 (UTC)

Prime counting function
The Prime-counting function gives the number of primes less than 1024 as	18,435,599,767,349,200,867,866. How do we know this number so accurately? Do we know all the primes up-to this number?--Salix (talk): 12:59, 7 November 2012 (UTC)
 * If you continue reading the article, in the very next section you’ll learn that there are algorithms to compute the number of primes faster than listing them.—Emil J. 13:29, 7 November 2012 (UTC)
 * ... And as for the specific value of π(1024), references are given below the table: the method used in the (second, unconditional) computation is described in detail in .—Emil J. 13:38, 7 November 2012 (UTC)


 * Hidden in this question seems to be the question of whether 18,435,599,767,349,200,867,866 is an exact value. Is it? --91.120.48.242 (talk) 15:07, 7 November 2012 (UTC)
 * The value of the Prime-counting function for a given input is, by definition, exact. And as is contended in the link from the article EmilJ gives above, π(1024) = 18,435,599,767,349,200,867,866. I don't know if that's helpful to your question, though. Are you thinking of the logarithmic integral which approximates the prime-counting function? 142.94.236.112 (talk) 23:28, 7 November 2012 (UTC)
 * "It is contended" - so, I guess the OP (I'm not the OP, I'm the 91.120 you just replied to) is asking, "how do we know?" Your answer is helpful in saying that "it ought to be", i.e. by definition, to be correct.  That answers a portion of my question but not whether 18,435,599,767,349,200,867,866 is, in fact, correct, and if so, how it was arrived at.   (For what it's worth, yes, I personally thought it could very well be the result of a function that has a very low margin of error but does not necessarily get the last couple of digits exactly right.  Still, the digits as presented could be the "most right", based on the function i.e. on average the most correct, which it why it wouldn't have been rounded.) --91.120.48.242 (talk) 07:11, 8 November 2012 (UTC)
 * Fair questions! in reading the paper posted above, the author describes his method such that "The result of the computation, after adding in the various error terms, was an interval straddling a single integer". And this value matches a previous result which was weaker in the sense it assumed the Riemann Hypothesis, if that lends some credibility. The details of his method are in the paper of course, relying on many previous results. I'm sure there's an appropriate article discussing the epistemological issues of peer review and any basis of "mathematical truth" -- which the OP's question might lead one to ponder. 142.94.236.112 (talk) 14:49, 8 November 2012 (UTC)


 * For comparison, the Goldbach conjecture verification project has computed all primes below 4×1018. This is probably the highest limit for which all primes have been computed. There are 95,676,260,903,887,607 primes up to that. They were not stored but only existed briefly in ram. Small primes like this can be computed about as quickly as they can be read from a disk, and noone bothers to store trillions of them. PrimeHunter (talk) 16:58, 8 November 2012 (UTC)