cp's OEIS Frontend

A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

%I A056993 #69 May 25 2023 11:01:13
%S A056993 2,2,2,2,2,30,102,120,278,46,824,150,1534,30406,67234,70906,48594,
%T A056993 62722,24518,75898,919444
%N A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.
%C A056993 Smallest base value yielding generalized Fermat primes. - _Hugo Pfoertner_, Jul 01 2003
%C A056993 The first 5 terms correspond with the known (ordinary) Fermat primes. A probable candidate for the next entry is 62722^131072+1, discovered by Michael Angel in 2003. It has 628808 decimal digits. - _Hugo Pfoertner_, Jul 01 2003
%C A056993 For any n, a(n+1) >= sqrt(a(n)), because k^(2^(n+1))+1 = (k^2)^(2^n)+1. - _Jeppe Stig Nielsen_, Sep 16 2015
%C A056993 Does the sequence contain any perfect squares? If a(n) is a perfect square, then a(n+1) = sqrt(a(n)). - _Jeppe Stig Nielsen_, Sep 16 2015
%C A056993 If for a particular n, a(n) exists, then a(i) exist for all i=0,1,2,...,n. No proof is known that this sequence is infinite. Such a result would clearly imply the infinitude of A002496. - _Jeppe Stig Nielsen_, Sep 18 2015
%C A056993 919444 is a candidate for a(20). See Zimmermann link. -  _Serge Batalov_, Sep 02 2017
%C A056993 Now PrimeGrid has tested and double checked all b^(2^20) + 1 with b < 919444, so we have proof that a(20) = 919444. - _Jeppe Stig Nielsen_, Dec 30 2017
%H A056993 Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/primes/gfn.html">Generalized Fermat Prime Search</a>
%H A056993 Lucile and Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/primes/results.html">Generalized Fermat Prime Search</a>
%H A056993 Michael Goetz, <a href="https://t5k.org/primes/page.php?id=103235">id=103235 of Top 5000 Primes</a>
%H A056993 Luke Harmon, Gaetan Delavignette, Arnab Roy, and David Silva, <a href="https://eprint.iacr.org/2023/700">PIE: p-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption</a>, Cryptology ePrint Archive 2023/700.
%H A056993 Stephen Scott, <a href="https://t5k.org/primes/page.php?id=84401">id=84401 of Top 5000 Primes</a>
%H A056993 Sylvanus A. Zimmerman, <a href="http://www.primegrid.com/download/GFN-919444_1048576.pdf">PrimeGrid’s Generalized Fermat Prime Search</a>
%F A056993 a(n) = A085398(2^(n+1)). - _Jianing Song_, Jun 13 2022
%e A056993 The primes are 2^(2^0) + 1 = 3, 2^(2^1) + 1 = 5, 2^(2^2) + 1 = 17, 2^(2^3) + 1 = 257, 2^(2^4) + 1 = 65537, 30^(2^5) + 1, 102^(2^6) + 1, ....
%t A056993 f[n_] := (p = 2^n; k = 2; While[cp = k^p + 1; !PrimeQ@cp, k++ ]; k); Do[ Print[{n, f@n}], {n, 0, 17}] (* _Lei Zhou_, Feb 21 2005 *)
%o A056993 (PARI) a(n)=my(k=2);while(!isprime(k^(2^n)+1),k++);k \\ _Anders Hellström_, Sep 16 2015
%Y A056993 Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.
%Y A056993 Cf. A019434 (Fermat primes).
%K A056993 hard,more,nonn
%O A056993 0,1
%A A056993 _Robert G. Wilson v_, Sep 06 2000
%E A056993 1534 from _Robert G. Wilson v_, Oct 30 2000
%E A056993 62722 from _Jeppe Stig Nielsen_, Aug 07 2005
%E A056993 24518 and 75898 from _Lei Zhou_, Feb 01 2012
%E A056993 919444 from _Jeppe Stig Nielsen_, Dec 30 2017