A037896 Primes of the form k^4 + 1.

2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001

Offset

1,1

Comments

From Bernard Schott, Apr 22 2019: (Start)

These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a perfect biquadrate: A307690, so this sequence is a subsequence of A078164 and A307690.

If p prime = k^4 + 1, phi(p) = k^4.

The last three Fermat primes in A019434 {17, 257, 65537} belong to this sequence; with F_k = 2^(2^k) + 1 and for k = 2, 3, 4, phi(F_k) = (2^(2^(k-2)))^4. (End)

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

M. Lal, Primes of the form n^4 + 1, Math. Comp. 21 (1967), pp. 245-247.

Example

6^4 + 1 = 1297 is prime.

Mathematica

Select[Range[200]^4+1, PrimeQ] (* Harvey P. Dale, Jul 20 2015 *)

Prog

(PARI) j=[]; for(n=1, 200, if(isprime(n^4+1), j=concat(j, n^4+1))); j
(PARI) list(lim)=my(v=List([2]), p); forstep(k=2, sqrtnint(lim\1-1, 4), 2, if(isprime(p=k^4+1), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Mar 31 2022
(Magma) [n^4+1: n in [1..200] | IsPrime(n^4+1)]; // G. C. Greubel, Apr 28 2019
(Sage) [n^4+1 for n in (1..200) if is_prime(n^4+1)] # G. C. Greubel, Apr 28 2019

Crossrefs

Cf. A000068, A002523, A019434.

Subsequence of A002496, A078164 and A039770.

Sequence in context: A002590 A029735 A291206 * A099714 A301584 A195443

Adjacent sequences: A037893 A037894 A037895 * A037897 A037898 A037899

Keyword

easy,nonn

Author

Donald S. McDonald, Feb 27 2000

Extensions

Corrected and extended by Jason Earls, Jul 19 2001

Status

approved