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

%I

%S 22,127,55,1527,18453,5517

%N The least integer k > 1 such that 1 + k^(5^n) + k^(2*5^n) + k^(3*5n) + k^(4*5^n) is prime.

%C Phi[5^(n+1),k]=1+k^(5^n)+k^(2*5^n)+k^(3*5^n)+k^(4*5^n)

%C 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.

%C 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.

%C 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.

%H David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/chgcertd.gp">Coppersmith--Howgrave-Graham certificate tester</a> (2006)

%H Chris Caldwell, <a href="http://primes.utm.edu/bios/page.php?id=797">John Renze's Coppersmith-Howgrave-Graham PARI script</a>

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

%t 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}]

%o (PARI) See Broadhurst link.

%o (PARI) a(n)=my(k=2);while(!ispseudoprime(polcyclo(5,k^n)),k++);k \\ _Charles R Greathouse IV_, Feb 09 2012

%Y Cf. A153438.

%K nonn,hard

%O 1,1

%A _Lei Zhou_, Feb 09 2012

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Last modified March 8 11:10 EST 2021. Contains 341948 sequences. (Running on oeis4.)