A153438 -id:A153438 - cp's OEIS frontend

A206418 a(n) is the least integer k > 1 such that 1 + k^(5^n) + k^(25^n) + k^(35^n) + k^(4*5^n) is prime.

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

Offset: 0

Lei Zhou, Feb 09 2012

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.

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

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

Cf. A085398, A153438.

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

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

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

A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).

Original entry on oeis.org

2, 4, 2, 6, 2, 20, 20, 26, 25, 10, 14, 5, 373, 4, 65, 232, 56, 2, 521, 911, 1156, 1619, 647, 511, 34, 2336, 2123, 1274, 2866, 951, 2199, 1353, 4965, 7396, 13513, 3692, 14103, 32275, 2257, 86, 3928, 2779, 18781, 85835, 820, 16647, 2468, 26677, 1172, 38361, 40842

Offset: 1

Lei Zhou, Apr 04 2012

1 - m^k + m^(2*k) - m^(3^k) + m^(4*k) equals Phi(10*k,m).
First 15 terms were generated by the provided Mathematica program. All other terms found using OpenPFGW as Fermat and Lucas PRP. Term 16-20, 22-24, 27 have N^2-1 factored over 33.3% and proved using OpenPFGW;
terms 21, 25, 29-33, 36, 37, 39, 41, 42, 45, 48, 51 are proved using CHG pari script;
terms 26, 28, 34, 40 are proved using kppm PARI script;
terms 35, 38, 43, 44, 46, 47, 49, 50 do not yet have a primality certificate.
The corresponding prime number of term 51 (40842) has 236089 digits.
The corresponding prime numbers for the following terms are equal:
  p(3)  = p(2)  = Phi(10, 2^4),
  p(12) = p(9)  = Phi(10, 5^50),
  p(18) = p(14) = Phi(10, 2^160),
  p(25) = p(21) = Phi(10, 34^512),
  p(40) = p(34) = Phi(10, 86^4000).

			n=1, A003592(1) = 1, when a=2, 1 - 2^1 + 2^2 - 2^3 + 2^4 = 11 is prime, so a(1)=2;
n=2, A003592(2) = 2, when a=4, 1 - 4^2 + 4^4 - 4^6 + 4^8 = 61681 is prime, so a(2)=4;
...
n=13, A003592(13) = 64, when a=373, PrimeQ(1 - 373^64 + 373^128 - 373^192 + 373^256) = True, while for a = 2..372, PrimeQ(1 - a^64 + a^128 - a^192 + a^256) = False, so a(13)=373.

Lei Zhou, Prime certificates of the corresponding primes of this sequence, April 2012.

Cf. A003592, A085398, A153438, A205506, A206418.

fQ[n_] := PowerMod[10, n, n] == 0;a = Select[10 Range@100, fQ]/10; l = Length[a]; Table[m = a[[j]]; i = 1;While[i++; cp = 1 - i^m + i^(2*m)-i^(3*m)+i^(4*m); ! PrimeQ[cp]]; i, {j, 1, l}]

do(k)=my(m=1);while(!ispseudoprime(polcyclo(10*k,m++)),);m
list(lim)=my(v=List(), N); for(n=0, log(lim)\log(5), N=5^n; while(N<=lim, listput(v, N); N<<=1)); apply(do, vecsort(Vec(v))) \\ Charles R Greathouse IV, Apr 04 2012

a(n) = A085398(10*A003592(n)). - Jinyuan Wang, Jan 01 2023

Term 50 added and comments updated by Lei Zhou, Jul 27 2012

Term 51 added and comments updated by Lei Zhou, Oct 10 2012

A298206 a(n) is the smallest b >= 2 such that b^(62^n) - b^(32^n) + 1 is prime.

Original entry on oeis.org

6, 3, 3, 6, 5, 106, 207, 569, 224, 736, 2854, 21234, 14837, 165394, 24743, 62721, 237804, 143332

Offset: 0

Serge Batalov, Jan 14 2018

a(13) = 165394 is a significant outlier from the generally expected trend, which can be conjectured to be 6*2^n*gamma, where gamma is the Euler-Mascheroni constant A001620. Additionally, the next b > a(13) such that b^(6*2^n) - b^(3*2^n) + 1 is prime is 165836, which is remarkably close to a(13). - Serge Batalov, Jan 24 2018

			2^12 - 2^6 + 1 = 4033 is composite and 3^12 - 3^6 + 1 = 530713 is prime, so a(1) = 3.

Phil Carmody, Prime Internet Eisenstein Search (ca. 2004-2005)
Mersenneforum, Prime Internet Eisenstein Search discussion
The Prime Pages, Generalized unique primes

Subsequence of A205506.

Cf. A001620, A085398, A153438, A246119.

for(n=0,9,for(b=2,1000,x=b^(3*2^n); if(isprime(x*(x-1)+1), print1(b,", "); break)))

a(n) = A085398(18*2^n). - Jinyuan Wang, Dec 21 2022

a(13) from Serge Batalov, Jan 24 2018

cp's OEIS Frontend

A206418 a(n) is the least integer k > 1 such that 1 + k^(5^n) + k^(25^n) + k^(35^n) + k^(4*5^n) is prime.

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A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).

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A298206 a(n) is the smallest b >= 2 such that b^(62^n) - b^(32^n) + 1 is prime.

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cp's OEIS Frontend

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.

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A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).

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A298206 a(n) is the smallest b >= 2 such that b^(6*2^n) - b^(3*2^n) + 1 is prime.

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A206418 a(n) is the least integer k > 1 such that 1 + k^(5^n) + k^(25^n) + k^(35^n) + k^(4*5^n) is prime.

A298206 a(n) is the smallest b >= 2 such that b^(62^n) - b^(32^n) + 1 is prime.