# Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ – are there bigger examples known?

I’m rereading an older text on fermat-quotients (see wikipedia) from which I have now the

Question for
$$b^{p-1} \equiv 1 \pmod{ p^m} \qquad \text{ with p \in \mathbb P , 1 \lt b \lt p and m \gt 2}$$
(This is a generalization of the question for Wieferich primes).

Note that I ask here for examples, where the bases $b$ are smaller than the prime $p$, so a very well known weaker case $3^{10} \equiv 1 \pmod {11^2 }$ were an example, but only if the exponent at $11$ where one more; however frequent and well known cases like $18^6 \equiv 1 \pmod {7^3}$ were not because the base is bigger than the prime.

The only example that I’ve found so far is
$$68^{112} \equiv 1 \pmod {113^3 }$$
but I’ve scanned only the first 2000 primes $p \in (3 \ldots 17389)$ and my primitive brute force algorithm has more than quadratic time-characteristic, so checking 10 000 or 100 000 primes were no fun – the quadratic regression prognoses 1 hour for testing 10 000 primes and 101 hours for testing 100 000 primes…

I’m aware of a couple of webpages containing lists of fermat quotients up to much higher primes, but either there is no explicite mention of the cases of $b \lt p$ and quotient $m \gt 2$ or I’ve been too dense when scanning through the listings (Richard Fischer, Wilfrid Keller, Michael Mossinghoff)

For reference: my Pari/GP-code is

for(j=2,2000,p=prime(j);p3=p^3;
for(k=2,p-1,
r = lift(Mod(k,p3)^(p-1));
if(r==1,print(p," ",k," ",r)));
);


P.s. I’ve no real good idea for tagging of this question; I just tried the most similar…

#### Solutions Collecting From Web of "Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ – are there bigger examples known?"

In MO there was an answer indicating, that there shall be no more information than that of Richard Fischer’s site, where he lists, that indeed that pair $(68,113)$ is the only pair up to about $p \le 3.6 \cdot 10^6$ and where also $b \lt p$ which gives a fermat-quotient greater than 2 , so I think I should “close the case” here.

For the casual reader I’ll add a link to a more explanative description of the problem and my empirical table. See here.