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After further thought, I realized that most of these congruences are all immediate consequences of the well-known property of Fermat quotients $$ q_p (a^n) \equiv n q_p (a) \pmod{p} $$ along with the …

In an answer to my own question, I showed that if $p$ is an odd prime and $p=a^m \pm b^n$ with positive integers $a,b$ relatively prime to $p$, then $p$ does not simultaneously divide the Fermat quoti …

In responding to Fast computation of multiplicative inverse modulo q I mentioned an algorithm for computing the inverse of $a \mod p$ different from the extended Euclidean algorithm, hoping that so …

This a repost of a question I asked at Stack Exchange: https://math.stackexchange.com/questions/35264/congruences-for-fermat-quotients I didn't get a complete answer to my question, so I'm trying …

(cross-posted from Math SE) For a given integer $b \geq 1$, let $D_b(m)$ be the number of positive divisors of $m$ that are less than $b$, and let $d_b(m)$ be the number of divisors of $m$ that are c …

By "Shintani domain", I mean a fundamental domain for the action of the totally positive units of a totally real number field k with $[k \colon \mathbb{Q}]=n$ (or more generally, those congruent to 1 …

Instead of going all the way to the GCD with the Euclidean algorithm and working backwards to find a multiplicative inverse, you can go straight to the multiplicative inverse with the Euclidean algori …

Many have suggested that it comes from "Lejeune", as in "Johann Peter Gustav Lejeune Dirichlet". I have never seen this properly sourced and have often wondered if the claim is legitimate.

In his preface to "Rapport sur la Cohomologie des Groups", Serge Lang says that those notes "provided missing chapters to the Artin-Tate notes on class field theory". It is available in english trans …

If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$ --- let's also assume $K/k$ is not generated by $p$-power roots of unity), I …

(Cross-posted from stackexchange: https://math.stackexchange.com/questions/1976296/what-is-known-about-the-average-of-the-partial-quotients-in-the-fundamental-peri) I'd like references concerning th …

(Cross-posted from https://math.stackexchange.com/questions/2029407/ray-class-groups-through-binary-quadratic-forms) If $d$ is the discriminant of a quadratic number field, then the primitive classes …

For each prime number $p$ and number field $k$, there exists at least one extension $k_{\infty}/k$ with Galois group isomorphic to $\mathbb{Z}_p$, the cyclotomic $\mathbb{Z}_p$-extension. If $k_p/k$ …

I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those tha …

I am getting ready to publish the manuscript http://arxiv.org/pdf/1408.4631v2.pdf and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from Name …