Smallest prime of the form a^2^m + b^2^m, m>=14

[JeppeSN]

Originally Posted by axn

(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

Submitted as A302387 (not approved yet). /JeppeSN

[JeppeSN]

Originally Posted by Batalov

Aw well, m=16 was too easy. Even a pair for the good measure:
124^65536+57^65536
143^65536+106^65536

I just discovered these are duplicates at the PRP Top site. If you search for x^(2^16)+y^(2^16), the same two numbers were found by Valeryi Kuryshev in Feb 2011.

/JeppeSN

[Batalov]

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 should better have 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]

OK. And still no trace of A291944(17), the smallest PRP of form xGF(17, a, b) = a^131072 + b^131072. /JeppeSN

[Batalov]

How far did anybody search?

[JeppeSN]

Kellen Shenton found the PRP 1196^131072 + 595^131072, to appear on PRP Top.

As soon as it is verified that 1196 is minimal for this exponent, OEIS will be updated.

/JeppeSN

[JeppeSN]

Ryan Propper found PRPs (35963^524288+1)/2 and (187503^262144+1)/2 and added them to A275530. /JeppeSN

[JeppeSN]

Kellen Shenton found PRP 200^262144 + 119^262144, and A291944 is updated. /JeppeSN