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Primes of the form k^4 + 1.
33

%I #54 Sep 08 2022 08:44:53

%S 2,17,257,1297,65537,160001,331777,614657,1336337,4477457,5308417,

%T 8503057,9834497,29986577,40960001,45212177,59969537,65610001,

%U 126247697,193877777,303595777,384160001,406586897,562448657,655360001

%N Primes of the form k^4 + 1.

%C From _Bernard Schott_, Apr 22 2019: (Start)

%C 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.

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

%C 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)

%H T. D. Noe, <a href="/A037896/b037896.txt">Table of n, a(n) for n=1..1000</a>

%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

%H M. Lal, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0222007-9">Primes of the form n^4 + 1</a>, Math. Comp. 21 (1967), pp. 245-247.

%e 6^4 + 1 = 1297 is prime.

%t Select[Range[200]^4+1,PrimeQ] (* _Harvey P. Dale_, Jul 20 2015 *)

%o (PARI) j=[]; for(n=1,200, if(isprime(n^4+1),j=concat(j,n^4+1))); j

%o (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

%o (Magma) [n^4+1: n in [1..200] | IsPrime(n^4+1)]; // _G. C. Greubel_, Apr 28 2019

%o (Sage) [n^4+1 for n in (1..200) if is_prime(n^4+1)] # _G. C. Greubel_, Apr 28 2019

%Y Cf. A000068, A002523, A019434.

%Y Subsequence of A002496, A078164 and A039770.

%K easy,nonn

%O 1,1

%A _Donald S. McDonald_, Feb 27 2000

%E Corrected and extended by _Jason Earls_, Jul 19 2001