Primes that are the sum of a positive cube and a square in 13 ways < 107122676734733201 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:17:32Z http://mathoverflow.net/feeds/question/103905 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103905/primes-that-are-the-sum-of-a-positive-cube-and-a-square-in-13-ways-107122676734 Primes that are the sum of a positive cube and a square in 13 ways < 107122676734733201 Kevin Acres 2012-08-04T00:04:59Z 2012-09-24T04:21:05Z <p>In order to try and add to the integer sequence at <a href="http://oeis.org/A173795" rel="nofollow">http://oeis.org/A173795</a> I am attempting to fill in a gap in a sequence of primes that are the sum of a positive cube and a square in n different ways.</p> <p>To date, and as far as I am aware, for n=0 to 10 the smallest primes are known that fulfill this criteria. The largest being 333413867957257 where n = 10.</p> <p>Candidates for the smallest at n=11 and n=12 are due to Elkies and these are 4417190430889897 and 84658174289284249 respectively.</p> <p>There then exists a gap at n=13 before 107122676734733201 fulfills the criteria for n=14.</p> <p>My questions are:</p> <ol> <li>Are 4417190430889897 and 84658174289284249 the smallest primes for n=11 and 12 respectively?</li> <li>Is 107122676734733201 the smallest prime where n=14?</li> <li>Is there a known prime &lt; 107122676734733201 where n=13?</li> </ol> <p>Kevin.</p> http://mathoverflow.net/questions/103905/primes-that-are-the-sum-of-a-positive-cube-and-a-square-in-13-ways-107122676734/107736#107736 Answer by Kevin Acres for Primes that are the sum of a positive cube and a square in 13 ways < 107122676734733201 Kevin Acres 2012-09-21T04:46:59Z 2012-09-24T04:21:05Z <p>Many thanks to Noam Elkies for taking the time and effort to resolve the first part of this question and for sending me the results by email.</p> <p>His, summarized, answer is:</p> <p>The smallest primes for \$n=11\$ and \$n=12\$ are respectively \$p=1057543811051633\$ and \$p=1448734752622601\$.</p> <p>For \$(n, p) = (12, 1448734752622601)\$. There are no other examples of \$n>11\$ for prime \$p\$ less than \$7.5 * 10^{15}\$.</p>