The "universal" diophantine equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:34:41Z http://mathoverflow.net/feeds/question/81427 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81427/the-universal-diophantine-equation The "universal" diophantine equation afexresearch 2011-11-20T14:54:30Z 2011-11-20T15:18:14Z <p>There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent </p> <ol> <li>All other diophantine equations (could be wrong on this)</li> <li>Any particular set of numbers -- such as the primes</li> </ol> <p>So to ask some questions around the consequence of this fact with another fact : the non-existence of a universal procedure for solving any diophantine. </p> <p>Since no such procedure can exist, am I correct in concluding from these two facts that, at least as represented by a diophantine set, the primes can not be enumerated?</p> <p>Or is the caveat that, maybe a particular procedure for solving a class or case of diophantine equations, of which the universal one mentioned above could be a member, exists and thus the primes could be enumerated by solving the universal one using such a yet-to-be-discovered specific method. </p> <p>Also, one final question: Am I right in my feeling that construction of this universal diophantine is not really "adding any new insight" to the area of primes, but simply finding a way to represent some kind of computer or turing machine as a diophantine and program it.</p> <p>If anyone would be so kind as to offer a simple explanation of the specific method of "programming" this diophantine or the constraints that actually give rise to this "universal" diophantine being able to encode the set of the primes, I would be grateful. </p> http://mathoverflow.net/questions/81427/the-universal-diophantine-equation/81428#81428 Answer by Charles for The "universal" diophantine equation Charles 2011-11-20T15:18:14Z 2011-11-20T15:18:14Z <p>Any Diophantine set can be enumerated, in the sense that there is a procedure that will list any given member of the set after a finite amount of time. In fact, Diophantine sets are precisely those which can be so listed: the recursively enumerable sets.</p> <p>For the primes and many other Diophantine sets, more is true: the elements can be listed in order, since there are procedures for determining not only existence but also non-existence. These sets are called <em>recursive</em> or <em>decidable</em>.</p> <p>You are right that finding a Diophantine representation for primes does not add much if anything to the study of primes. It serves, rather, to increase our understanding of Diophantine equations.</p>