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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
22, 127, 55, 1527, 18453, 5517 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..6.

David Broadhurst, Coppersmith--Howgrave-Graham certificate tester (2006)

Chris Caldwell, John Renze's Coppersmith-Howgrave-Graham PARI script

EXAMPLE

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

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}]

PROG

(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

CROSSREFS

Cf. A153438.

Sequence in context: A252835 A220622 A229375 * A309923 A215626 A125247

Adjacent sequences:  A206415 A206416 A206417 * A206419 A206420 A206421

KEYWORD

nonn,hard

AUTHOR

Lei Zhou, Feb 09 2012

STATUS

approved

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Last modified January 17 04:44 EST 2021. Contains 340214 sequences. (Running on oeis4.)