A351865 Primes of the form x^2 + 64*y^2 that divide some Fermat number.

257, 65537, 2424833, 26017793, 63766529, 825753601, 1214251009, 6487031809, 2710954639361, 2748779069441, 6597069766657, 25991531462657, 76861124116481, 151413703311361, 1095981164658689, 1238926361552897, 1529992420282859521, 2663848877152141313, 3603109844542291969

Offset: 1

Author

Arkadiusz Wesolowski, Apr 10 2022

Keywords

• nonn

Comments

A prime p = k*2^j + 1 (with k odd) belongs to this sequence if and only if p is a factor of a Fermat number 2^(2^m) + 1 for some m <= j - 3.

Examples

			a(1) = 1^2 + 64*2^2 = 257 is a prime factor of 2^(2^3) + 1;
a(2) = 1^2 + 64*32^2 = 65537 is a prime factor of 2^(2^4) + 1;
a(3) = 127^2 + 64*194^2 = 2424833 is a prime factor of 2^(2^9) + 1;
a(4) = 2047^2 + 64*584^2 = 26017793 is a prime factor of 2^(2^12) + 1;
a(5) = 7295^2 + 64*406^2 = 63766529 is a prime factor of 2^(2^12) + 1;
		

References

• Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Links

• Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Crossrefs

Cf. A014754, A023394, A228846.

Programs

PARI

isok(p) = if(p%8==1 && isprime(p), my(d=Mod(2, p)); d^((p-1)/4)==1 && d^2^valuation(p-1, 2)==1, return(0));

Formula

A014754 INTERSECT A023394.