*(I've taken this from [MSE](https://math.stackexchange.com/questions/594708), it seems to be more appropriate here)*


I'm rereading an older text on fermat-quotients (see [wikipedia](https://en.wikipedia.org/wiki/Fermat_quotient)) 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](http://www.fermatquotient.com/), [Wilfrid Keller](http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html), [Michael Mossinghoff](http://www.cecm.sfu.ca/~mjm/WieferichBarker/))




---


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 $p=104729$ )*


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[update] For the later casual reader I've included a more involved explanation and a table of data. See [here](http://go.helms-net.de/math/expdioph/fermatquot_ge2_table1.htm) (filesize 2Mb, inconvenient for modem-transfer)



Fermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?
Cases with $b<p$ can be easily identified in Fischer's table <http://fermatquotient.com/FermatQuotienten/FermatQ3> and there are no other such instances for primes below $3.6\cdot 10^8$.


