A130076 - OEIS

%I #23 Feb 25 2023 03:05:51

%S 2,3,5,19

%N Primes p such that p^2 divides 5^p - 3^p - 2^p.

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

%C 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).

%C 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.

%C No other terms below 10^11. - _Max Alekseyev_, Dec 06 2010

%e 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}.

%t 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 *)

%o (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

%Y Cf. A130072, A130073, A130074, A130075.

%K bref,hard,more,nonn

%O 1,1

%A _Alexander Adamchuk_, May 06 2007

%E Edited by _Max Alekseyev_, Dec 05 2010