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In order to try and add to the integer sequence at 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.

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.

Candidates for the smallest at n=11 and n=12 are due to Elkies and these are 4417190430889897 and 84658174289284249 respectively.

There then exists a gap at n=13 before 107122676734733201 fulfills the criteria for n=14.

My questions are:

  1. Are 4417190430889897 and 84658174289284249 the smallest primes for n=11 and 12 respectively?
  2. Is 107122676734733201 the smallest prime where n=14?
  3. Is there a known prime < 107122676734733201 where n=13?


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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.

His, summarized, answer is:

The smallest primes for $n=11$ and $n=12$ are respectively $p=1057543811051633$ and $p=1448734752622601$.

For $(n, p) = (12, 1448734752622601)$. There are no other examples of $n>11$ for prime $p$ less than $7.5 * 10^{15}$.

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Has OEIS been notified? – Gerry Myerson Sep 21 '12 at 5:40
@Gerry - I've just updated the OEIS entry. – Kevin Acres Sep 22 '12 at 4:34
The updates are now showing at – Kevin Acres Sep 22 '12 at 22:08

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