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

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

Author

Lei Zhou, Apr 04 2012

Keywords

• nonn
• hard

Comments

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).

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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}]

PARI

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

Formula

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

Extensions

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

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