Intereting Posts

Combinatorial Proof: $n! = 1+\sum\limits_{r=1}^{n-1}( r\cdot r!)$
Essential ideals
Finding an example of a non-rational p-adic number
Uniform continuity on (0,1) implies boundedness
Proving the converse of an IMO problem: are there infinitely many pairs of positive integers (m,n) such that m divides n^2 + 1 and n divides m^2 + 1?
Complex analysis prerequisites
programming language with HALTS keyword
If $M\oplus M$ is free, is $M$ free?
Zero divisors in $A$
$p \leqslant q \leqslant r$. If $f \in L^p$ and $f \in L^r$ then $ f \in L^q$?
how to prove $\sum_{k=0}^{m}\binom{n+k}{n}=\binom{n+m+1}{n+1}$ without induction?
Proof that the series expansion for exp(1) is a Cauchy sequence
How prove this $\displaystyle\lim_{n\to \infty}\frac{n}{\ln{(\ln{n}})}\left(1-a_{n}-\frac{n}{\ln{n}}\right)=-1$
Integrating a product of exponentials and error functions
Intuition behind Taylor/Maclaurin Series

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.

- How to properly set up partial fractions for repeated denominator factors
- Equivalence of Quadratic Forms that represent the same values
- On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
- Integral solutions to $y^{2}=x^{3}-1$
- a new continued fraction for $\sqrt{2}$
- What is the inverse of the Carmichael-function?

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…

- $\Delta^ny = n!$ , difference operator question.
- Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?
- Positive Integers Equation
- Are limits on exponents in moduli possible?
- Is there any real number except 1 which is equal to its own irrationality measure?
- Positive integer solutions to $\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016^m$
- Smallest number with a given number of factors
- Efficiently calculating the logarithmic integral with complex argument
- Given the Cauchy's problem: $y'' = 1, y(0) = 0, y'(0) = 0$. Why finite difference method doesn't agree with recurrence equation?
- Can the Basel problem be solved by Leibniz today?

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.

- Extension of bounded convex function to boundary
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- Are there any elegant methods to classify of the Gaussian primes?
- Proving that the medians of a triangle are concurrent
- If the minimal polynomial is irreducible $\bmod p$ for some $p$, is the Galois group cyclic?
- Using Burnside's Lemma; understanding the intuition and theory