OFFSET
0,1
COMMENTS
Smallest base value yielding generalized Fermat primes. - Hugo Pfoertner, Jul 01 2003
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
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
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
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
919444 is a candidate for a(20). See Zimmermann link. - Serge Batalov, Sep 02 2017
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
LINKS
Yves Gallot, Generalized Fermat Prime Search
Lucile and Yves Gallot, Generalized Fermat Prime Search
Michael Goetz, id=103235 of Top 5000 Primes
Luke Harmon, Gaetan Delavignette, Arnab Roy, and David Silva, PIE: p-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption, Cryptology ePrint Archive 2023/700.
I. Lunev, A tribonacci-like sequence of composite numbers, J. Int. Seq. 20 (2017) # 17.3.2
Stephen Scott, id=84401 of Top 5000 Primes
Sylvanus A. Zimmerman, PrimeGrid’s Generalized Fermat Prime Search
FORMULA
a(n) = A085398(2^(n+1)). - Jianing Song, Jun 13 2022
EXAMPLE
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, ....
MATHEMATICA
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 *)
PROG
(PARI) a(n)=my(k=2); while(!isprime(k^(2^n)+1), k++); k \\ Anders Hellström, Sep 16 2015
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Robert G. Wilson v, Sep 06 2000
EXTENSIONS
1534 from Robert G. Wilson v, Oct 30 2000
a(17) from Jeppe Stig Nielsen, Aug 07 2005
a(18)-a(19) from Lei Zhou, Feb 01 2012
a(20) from Jeppe Stig Nielsen, Dec 30 2017
a(21) from Jeppe Stig Nielsen, Dec 01 2025
STATUS
approved