A206418 a(n) is the least integer k > 1 such that 1 + k^(5^n) + k^(2*5^n) + k^(3*5^n) + k^(4*5^n) is prime.

2, 22, 127, 55, 1527, 18453, 5517

Offset: 0

Author

Lei Zhou, Feb 09 2012

Keywords

• nonn
• hard
• more

Comments

Phi(5^(n+1), k) = 1 + k^(5^n) + k^(2*5^n) + k^(3*5^n) + k^(4*5^n).

The primes correspond to k(1) through k(4) have a p-1 factorable up to 34% or higher, thus are proved prime by OpenPFGW.

The fifth one, Phi(5^6,18453) = 1 + 18453^3125 + 18453^6250 + 18453^9375 + 18453^12500, is a 55326-digit Fermat and Lucas PRP with 78.86% proof. A CHG proofing is running but it will take month to complete.

The sixth one, Phi(5^7,5517), has 233857 digits and can only be factored to about 26%. It is too big for CHG to provide a proof.

Examples

			Phi(5^2, 22) = 705429635566498619547944801 is prime, while Phi(25, k) with k = 2 to 21 are composites, so a(1) = 22.
		

Links

• David Broadhurst, Coppersmith--Howgrave-Graham certificate tester (2006)
• Chris Caldwell, John Renze's Coppersmith-Howgrave-Graham PARI script

Crossrefs

Cf. A085398, A153438.

Programs

Mathematica

Table[i = 1; m = 5^u; While[i++; cp = 1 + i^m + i^(2*m) + i^(3*m) + i^(4^m); ! PrimeQ[cp]]; i, {u, 1, 4}]

PARI

See Broadhurst link.

PARI

a(n)=my(k=2);while(!ispseudoprime(polcyclo(5,k^n)),k++);k \\ Charles R Greathouse IV, Feb 09 2012

Formula

a(n) = A085398(5^(n+1)). - Jinyuan Wang, Dec 21 2022

Extensions

a(0) inserted by Jinyuan Wang, Dec 21 2022