Intereting Posts

Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence
Prove Euler's Theorem when the integers are not relatively prime
Relation betwen coefficients and roots of a polynomial
How to efficiently solve a series of similar matrix equations using the LU decomposition
Correct way to calculate numeric derivative in discrete time?
Intuitive ways to get formula of cubic sum
Motivation behind the definition of ideal class group
What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?
What is a complex inner product space “really”?
Probabilistic proof of existence of an integer
Can a basis for a vector space be made up of matrices instead of vectors?
Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)
Prove the limit exists
Is there such a thing as a countable set with an uncountable subset?
Justify: In a metric space every bounded sequence has a convergent subsequence.

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.

- Efficient Method to find funsum$(n) \pmod m$ where funsum$(n)= 0^0 + 1^1 + 2^2 + … n^n$
- Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?
- A series with only rational terms for $\ln \ln 2$
- The Greatest Common Divisor of All Numbers of the Form $n^a-n^b$
- Prove that $\dfrac{\pi}{\phi^2}<\dfrac{6}5 $
- Integer solutions of $x^3+y^3=z^2$

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…

- Is anyone talking about “ball bundles” of metric spaces?
- Sum of greatest common divisors
- What is the intuition behind the concept of Tate twists?
- Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
- If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.
- Is there a conjecture with maximal prime gaps
- Find all integer solutions to Diophantine equation $x^3+y^3+z^3=w^3$
- Families of curves over number fields
- Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q \implies \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
- Can you determine a formula for this problem?

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.

- Maximum area of triangle inside a convex polygon
- How to choose between two options with a biased coin
- Minimum of $ f(\alpha) = \left(1+\frac{1}{\sin^{n}\alpha}\right)\cdot \left(1+\frac{1}{\cos^{n}\alpha}\right)$
- Trying to derive a contradiction with this simple inequality,
- $n$ lines cannot divide a plane region into $x$ regions, finding $x$ for $n$
- Equivalences of continuity, sequential convergence iff limit (S.A. pp 106 t4.2.3, 110 t4.3.2)
- Associativity of $\iff$
- Logic – how to write $\exists !x$ without the $\exists !$ symbol
- Is each edge interpreted like a $2$-cycle?
- Unexpected approximations which have led to important mathematical discoveries
- Property of sup of a set of numbers
- $Tf=\sum\limits_{n=1}^\infty f(n)x_n$ is surjective from $\ell^1$ to a separable Banach space
- prove that $(n^2)!$ is divisible by $(n!)^{n+1}$, where $n\in \mathbb{N}$
- Are there any sets other than the usual in which we can apply Sturm's axioms?
- Real Analysis Methodologies to show $\gamma =2\int_0^\infty \frac{\cos(x^2)-\cos(x)}{x}\,dx$