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A206418
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The least integer k > 1 such that 1 + k^(5^n) + k^(2*5^n) + k^(3*5n) + k^(4*5^n) is prime.
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2
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OFFSET
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1,1
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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=1..6.
David Broadhurst, Coppersmith--Howgrave-Graham certificate tester (2006)
Chris Caldwell, John Renze's Coppersmith-Howgrave-Graham PARI script
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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 k(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. A153438.
Sequence in context: A252835 A220622 A229375 * A309923 A215626 A125247
Adjacent sequences: A206415 A206416 A206417 * A206419 A206420 A206421
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KEYWORD
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nonn,hard
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AUTHOR
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Lei Zhou, Feb 09 2012
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STATUS
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approved
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