A253234 Smallest prime p such that p^2 divides n^(p-1)-(n-1)^(p-1).

1093, 23, 5, 3, 3457, 72673, 13, 67, 67

Offset: 2

Author

Eric Chen, May 17 2015

Keywords

• nonn
• more
• hard

Comments

According to the generalized Fermat's little theorem, if a prime p does not divide n or n-1, then p must divide n^(p-1)-(n-1)^(p-1), so we can ask that whether there exists a Wieferich-type prime to the "base" (n, n-1), that is, a prime p such that p^2 divides n^(p-1)-(n-1)^(p-1).

a(n) is currently unknown for n = {11, 36, 49, 52, 66, 70, 84, 89, 102, 112, 133, 142, 148, ...}

a(11) > 6.5*10^10.

Links

• Eric Chen, Table of n, a(n) for n = 2..1000 status (test limit: 6.5*10^10)
• Richard Fischer, Primes p such that (n+1)^(p-1)-n^(p-1) == 0 (mod p^2)

Crossrefs

Cf. A039951, A174422.

Programs

Mathematica

f[n_] := Block[{k = 1}, While[k < 10^8 && PowerMod[n, Prime[k] - 1, Prime[k]^2] != PowerMod[n - 1, Prime[k] - 1, Prime[k]^2], k++ ]; If[k == 10^8, 0, Prime[k]]]; Table[ f[n], {n, 2, 75}]

PARI

for(n=2, 10, forprime(p=2, 1e8, if(Mod(n, p^2)^(p-1)==Mod(n-1, p^2)^(p-1), print1(p, ", "); next({2}))); print1("--, "))

Formula

a(9k+5) = 3.