A130076 Primes p such that p^2 divides 5^p - 3^p - 2^p.

2, 3, 5, 19

Offset: 1

Author

Alexander Adamchuk, May 06 2007

Keywords

• bref
• hard
• more
• nonn

Comments

For a prime p, p divides A130072(p) = 5^p - 3^p - 2^p. Quotients A130072(p)/p are listed in A130075.

If p^2 divides A130072(p), then p^(k+1) divides A130072(p^k) for every k>0. For p = 19, even 19^(k+2) divides A130072(p^k).

Numbers n such that n divides A130072(n) are listed in A130073. Nonprimes n such that n divides A130072(n) are listed in A130074, which apparently include all powers p^k of primes p = {2,3,5,19} for k>1 and all powers of numbers of the form 2^k*3^m, 3^k*5^m, 5^k*19^m.

No other terms below 10^11. - Max Alekseyev, Dec 06 2010

Examples

			p^2 divides A130072(p) = 5^p - 3^p - 2^p for prime p = {2,3,5,19}, quotients A130072(p)/p^2 are {3,10,114,52831921170}.
		

Crossrefs

Cf. A130072, A130073, A130074, A130075.

Programs

Mathematica

fQ[p_]:=Mod[PowerMod[5,p,p^2]-PowerMod[3,p,p^2]-PowerMod[2,p,p^2],p^2]==0 (* Robert G. Wilson v, Mar 14 2011 *)

PARI

forprime(p=2,1e9,if(Mod(5,p^2)^p==Mod(3,p^2)^p+Mod(2,p^2)^p,print1(p", "))) \\ Charles R Greathouse IV, Mar 14 2011

Extensions

Edited by Max Alekseyev, Dec 05 2010