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-   -   Smallest prime of the form a^2^m + b^2^m, m>=14 (https://www.mersenneforum.org/showthread.php?t=23160)

 JeppeSN 2018-04-06 17:26

[QUOTE=axn;483415][CODE](3^2+1^2)/2
(3^4+1^4)/2
(5^8+3^8)/2
(3^16+1^16)/2
(3^32+1^32)/2
(3^64+1^64)/2
(179^128+177^128)/2
(169^256+167^256)/2
(935^512+933^512)/2
(663^1024+661^1024)/2[/CODE][/QUOTE]

Submitted as [URL="https://oeis.org/A302387"]A302387[/URL] (not approved yet). /JeppeSN

 JeppeSN 2018-12-15 23:24

[QUOTE=Batalov;482574]Aw well, m=16 was too easy. Even a pair for the good measure:
124^65536+57^65536
143^65536+106^65536[/QUOTE]

I just discovered these are duplicates at the PRP Top site. If you [URL="http://www.primenumbers.net/prptop/searchform.php?form=x%5E%282%5E16%29%2By%5E%282%5E16%29&action=Search"]search for x^(2^16)+y^(2^16)[/URL], the same two numbers were found by Valeryi Kuryshev in Feb 2011.

/JeppeSN

 Batalov 2018-12-16 00:17

Yes, there are many more; I've seen duplicates in other classes before. (Because there are similarly different ways to write values, e.g. using Phi(), or gcd() ... for primitive parts or Aurifeuillean parts. PRP owners [I]should better have[/I] an in-take evaluator and checker.

The bracketed form is hard to search for. You can use "x^?+y^?" and then filter, but that's not the first form that comes to mind.

 JeppeSN 2018-12-16 00:35

OK. And still no trace of [URL="https://oeis.org/A291944"]A291944[/URL](17), the smallest PRP of form xGF(17, a, b) = a^131072 + b^131072. /JeppeSN

 Batalov 2018-12-16 01:57

How far did anybody search?

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