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A206418
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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.
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2
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..6.
David Broadhurst, Coppersmith--Howgrave-Graham certificate tester (2006)
Chris Caldwell, John Renze's Coppersmith-Howgrave-Graham PARI script
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FORMULA
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a(n) = A085398(5^(n+1)). - Jinyuan Wang, Dec 21 2022
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EXAMPLE
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Phi(5^2, 22) = 705429635566498619547944801 is prime, while Phi(25, k) with k = 2 to 21 are composites, so a(1) = 22.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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Cf. A085398, A153438.
Sequence in context: A221697 A292452 A292732 * A255866 A330900 A356341
Adjacent sequences: A206415 A206416 A206417 * A206419 A206420 A206421
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KEYWORD
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nonn,hard,more,changed
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AUTHOR
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Lei Zhou, Feb 09 2012
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EXTENSIONS
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a(0) inserted by Jinyuan Wang, Dec 21 2022
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STATUS
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approved
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