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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). 1

%I

%S 2,4,2,6,2,20,20,26,25,10,14,5,373,4,65,232,56,2,521,911,1156,1619,

%T 647,511,34,2336,2123,1274,2866,951,2199,1353,4965,7396,13513,3692,

%U 14103,32275,2257,86,3928,2779,18781,85835,820,16647,2468,26677,1172,38361,40842

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

%C 1-m^k+m^(2*k)-m^(3^k)+m^(4*k) equals Phi(10*k,m).

%C 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;

%C Terms 21, 25, 29-33, 36, 37, 39, 41, 42, 45, 48, 51 are proved using CHG pari script;

%C Terms 26, 28, 34, 40 are proved using kppm pari script;

%C Terms 35, 38, 43, 44, 46, 47, 49, 50 do not yet have a primality certificate.

%C The corresponding prime number of term 51 (40842) has 236089 digits.

%C The corresponding prime numbers for the following terms are equal:

%C p(3) = p(2) = Phi[10, 2^4],

%C p(12) = p(9) = Phi[10, 5^50],

%C p(18) = p(14) = Phi[10, 2^160],

%C p(25) = p(21) = Phi[10, 34^512],

%C p(40) = p(34) = Phi[10, 86^4000].

%H Lei Zhou, <a href="http://bitc.bme.emory.edu/~lzhou/blogs/?p=263">Prime certificates of the corresponding primes of this sequence</a>.

%e n=1, A003592[1] = 1, when a=2, 1-2^1+2^2-2^3+2^4=11 is prime, so a(1)=2;

%e n=2, A003592[2] = 2, when a=4, 1-4^2+4^4-4^6+4^8=61681 is prime, so a(2)=4;

%e ...

%e 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;

%t 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}]

%o (PARI) do(k)=my(m=1);while(!ispseudoprime(polcyclo(10*k,m++)),);m

%o 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

%Y Cf. A003592, A205506, A153438, A206418.

%K nonn,hard

%O 1,1

%A _Lei Zhou_, Apr 04 2012

%E Added term 50 and updated comments - Lei Zhou, Jul 27 2012

%E Added term 51 and updated comments - Lei Zhou, Oct 10 2012

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Last modified February 26 13:59 EST 2021. Contains 341632 sequences. (Running on oeis4.)