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The search turned up a 1981 paper by John S.Lew (in the Unsolved problems section) Polynomials in Two Variables Taking Distinct Integer Values at Lattice-Points which discusses related problems, and …

Motivation I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form the set $\ …

$\DeclareMathOperator\PSL{PSL}$In the paper Ono - "Hasse principle" for $\PSL_2 (\mathbb Z)$ and $\PSL_2(\mathbb F)$ there's a definition of a Hasse principle for a group $G$, but I don't completely g …

I thought that the order of the Tate-Shafarevich group should always be a square (it's also supposed to be finite, but for the purposes of this question let's assume we know this) but I don't seem to …

I believe there was an old conjecture that there's always a prime number between N and 2N. What's the history and how is this proven is the easiest/elementary/deepest ways?

Trying to understand answer to this question. What is the (Beilinson) higher regulator of a number field?

I'm sure you omitted this just because it's too classic: big part of group theory was invented to prove that most algebraic numbers cannot be constructed by radical extensions. It's still the best, …

Source This question came up in the discussion between Kevin Buzzard and Minhyong Kim in the comments to Smooth proper scheme over Z. It was 2 weeks ago, so I took the liberty of posting it as commun …

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s. There's a very interesting article by Lan …

Along the lines of Polynomial representing all nonnegative integers, but likely well-known question: is there a polynomial $f \in \mathbb Q[x_1, \dots, x_n]$ such that $f(\mathbb Z\times\mathbb Z\ …

I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change f:Z\to Y, meaning that X \times_{Y} Z \to Z also has this property. Quest …

Are supersingular primes and supersingular elliptic curves related? (this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post …

Comments by Anweshi The essential point is what Emerton mentioned, ie the analogy with Minkowski's theorem on number fields with ramification. The basic principle is that "arithmetic is geometry". Nu …

What can be said about the polynomials $f\in\mathbb Q[x, y]$ which are nonnegative on $\mathbb R\times \mathbb R$? Motivation: this may lead to progress in the question about polynomial onto map …

That would be equivalent to $2(n-1) = k\varphi(n)$ and $n-1\ne k'\varphi(n)$ by Fermat's little theorem for composite numbers. The second condition is equivalent to being able to satisfy first with …