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, 723394817, 916636177, 1049760001, 1416468497

Offset: 1

Author

Donald S. McDonald, Feb 27 2000

Keywords

• easy
• nonn

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)

Examples

			6^4 + 1 = 1297 is prime.
		

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.

Crossrefs

Cf. A000068, A002523, A019434, A078164, A307690.

Subsequence of A002496, A078164 and A039770.

Programs

Magma

[n^4+1: n in [1..200] | IsPrime(n^4+1)]; // G. C. Greubel, Apr 28 2019

Mathematica

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

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

Sage

[n^4+1 for n in (1..200) if is_prime(n^4+1)] # G. C. Greubel, Apr 28 2019

Formula

a(n) = A002523(A000068(n)). - Elmo R. Oliveira, Feb 21 2025

Extensions

Corrected and extended by Jason Earls, Jul 19 2001