A096082 Smallest odd prime p such that p^2 | n^(p-1) - 1.

3, 1093, 11, 1093, 20771, 66161, 5, 3, 11, 3, 71, 2693, 863, 29, 29131, 1093, 3, 5, 3, 281

Offset: 1

Author

Lekraj Beedassy, Jul 22 2004

Keywords

• nonn
• more
• hard

Comments

Similar to the sequence A039951 where p=2 is allowed.

a(n^k) <= a(n) for any n,k>1.

a(21) > 1.63*10^14 (see Fischer's link).

For all nonnegative integers n and k, a(n^(n^k)) = a(n). (see puzzle 762 in the links). Also a(n) = 3 if and only if mod(n, 36) is in the set {1, 8, 10, 19, 26, 28, 35}. - Farideh Firoozbakht and Jahangeer Kholdi, Nov 01 2014

Links

• C. K. Caldwell, The Prime Glossary, Fermat quotient
• Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
• W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p
• Carlos Rivera, Puzzle 762. Conjecture from Ribenboim's book, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A007663, A001220, A039951, A124121, 124122.

Programs

Mathematica

f[n_] := Block[{k = 2}, While[k < 5181800 && PowerMod[n, Prime[k] - 1, Prime[k]^2] != 1, k++ ]; If[k == 5181800, 0, Prime[k]]]; Table[ f[n], {n, 70}] (* Robert G. Wilson v, Jul 23 2004 *)

PARI

for(n=2, 20, forprime(p=3, 1e9, if(Mod(n, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jul 24 2014

Formula

a(n) = A039951(n) for all n not of the form 4k+1, while a(4k+1) > A039951(4k+1) = 2. - Alexander Adamchuk, Dec 03 2006

Extensions

Definition corrected by Alexander Adamchuk, Nov 27 2006

Edited by Max Alekseyev, Oct 07 2009

Edited and updated by Max Alekseyev, Jan 29 2012