*(I've taken this from MSE, it seems to be more appropriate here)*

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)));
);
```

*(One correspondent took this up to the 10 000'th prime which is $p=104729$ )*

[update] For the later casual reader I've included a more involved explanation and a table of data. See here (filesize 2Mb, inconvenient for modem-transfer)