Is it true that for every integer $r$, the equation $2^m = 3^n + r$ has at most a finite number of integer solutions? I understand that this is a special case of Pillai's conjectur …

Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly …

Let $p$ be a prime. Given a positive integer $n$, does there exist a $\beta$ in an extension of $F_p$ such that 1) If $F_p[\beta] = F_{p^N}$, then $N > n^n$; ( $\beta$ lies in a …

Hello all, Dirichlet's famous theorem asserts that any arithmetic progression $\lbrace ax+b | x \in {\mathbb N}\rbrace$ contains infinitely many primes if a and b are relatively pr …

One of the reasons why the classical theory of binary quadratic forms is hardly known anymore is that it is roughly equivalent to the theory of ideals in quadratic orders. There i …

Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$? I gue …

Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$: (i) $p$ is inert in End($E$) (ii) $E_p$ is supersingular (iii) The trace of the Frobenius …

Given a set of distances S, choose N unique points P on a number line such that the distances between the N points occur in S as much as possible. That is, maximize the occurence i …

For a positive integer k, let d(k) be the number of divisors of k. So d(1)=1, d(p) =2 if p is a prime, d(6)=4, and d(12)=6. What is the precise asymptotics of SUM_{k=1}^n 1/(kd(k …

[Follow-up to Can every finite graph be represented by one prescribed sequence of natural numbers?, reformulated thanks to a hint from Jacques Carette] Let $V(n,\nu)$ and $E(n, …

I have two "vague questions" which are the following: if you have two $n$-dimensional $\ell$-adic Galois representations of a number field $K$ (with the standard ramification condi …

(This is a follow-up to my previous question Can every finite graph be represented by an arithmetic sequence of natural numbers?) Since it is obviously false that every finite gra …

Chevalley's theorem says that if $k$ is a finite field and $f(X_1,...,X_n)$ is a form (homogeneous polynomial) of degree $d < n$, then the equation $f(X_1,...,X_n) = 0$ has a no …

This is a question about terminology and should be taken lightly. The expression local field is used in at least three different senses : 1) For a locally compact totally discon …

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?