A188465 Primes p such that p^2 divides 2^(2^(p-1)-1) - 1.

7, 73, 127, 337, 487, 601, 881, 937, 1801, 2593, 2647, 3079, 3943, 4057, 4201, 6553, 7993, 9199, 10657, 14407, 15289, 16759, 18041, 18121, 20521, 20809, 21673, 22111, 24967, 25111, 26407, 28393, 28729, 36793, 39367, 41161, 42463, 47737, 47881, 49201, 49297

Offset: 1

Author

Arkadiusz Wesolowski, Apr 10 2011

Keywords

• nonn

Comments

Primes p that divide 2^(2^(p-1)-1) - 1 are the same terms. Proof: p | 2^(2^(p-1)-1) - 1 iff ord_{p}(2) | 2^(p-1)-1, so p^2 | 2^(2^(p-1)-1) - 1, since p | 2^(p-1)-1 by FLT. - Thomas Ordowski, Sep 16 2024

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Arkadiusz Wesolowski)

Crossrefs

Cf. A001220.

Programs

Maple

isA188465 := proc(p) local m; if isprime(p) then m := modp(2 &^ ( (2 ^ (p-1))-1)-1,p) ; m := simplify(m) ; if m = 0 then true; else false; end if; else false; end if; end proc:
for i from 1 do p := ithprime(i) ; if isA188465(p) then printf("%d\n",p) ; end if; end do: # R. J. Mathar, Apr 10 2011

Mathematica

okQ[p_] := Module[{k = MultiplicativeOrder[2, p^2]}, PowerMod[2, p - 1, k] == 1]; Select[Prime[Range[5000]], okQ] (* T. D. Noe, Apr 11 2011 *)