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Primes p such that (4^(p-1)-1) == 0 mod ((p-1)^2+1).
2

%I #15 Sep 08 2022 08:46:17

%S 3,5,17,257,8209,59141,65537,649801

%N Primes p such that (4^(p-1)-1) == 0 mod ((p-1)^2+1).

%C Prime terms from A273870.

%C The first 5 known Fermat primes from A019434 are in this sequence.

%C Conjecture 1: also primes p such that ((4^k)^(p-1)-1) == 0 mod ((p-1)^2+1) for all k >= 0.

%C Conjecture 2: supersequence of Fermat primes (A019434).

%e 5 is a term because (4^(5-1)-1) == 0 mod ((5-1)^2+1); 255 == 0 mod 17.

%o (Magma) [n: n in [1..100000] | IsPrime(n) and (4^(n-1)-1) mod ((n-1)^2+1) eq 0]

%o (PARI) is(n)=isprime(n) && Mod(4,(n-1)^2+1)^(n-1)==1 \\ _Charles R Greathouse IV_, Jun 08 2016

%Y Cf. A019434, A273870.

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Jun 01 2016