Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

10,454 questions
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Universality of $y^4-x^3$ mod $p$

For pedagogical reasons, I got interested in the equation $y^4-x^3=a$ over $\mathbf F_p$. To my surprise (maybe I'm naive), there is only one couple $(p,a)=(13,7)$ for which there is no solution, at ...
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Collection of equivalent forms of a precise statement: the number of elements of a well defined set is infinite [on hold]

This main aim of this forum is to collect a "big list" of equivalent forms of number theory problems equivalent to a statement of the form: The case (A) is true if and only if the number of elements ...
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Is there a conjectural analog of Ribet's theorem (Converse to Herbrand's Theorem) for Imaginary Quadratic fields?

For $p$ a prime, let $Cl(\mathbb{Q}(\mu_p))$ denote the class group of the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of unity. Associated to an eigenform of weight 2 and ...
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A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?

I have the following conjecture involving a possible new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$ with $p$ an odd prime. Conjecture. Let $p$ be an odd ...
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Question on the 50th (known) Mersenne prime number

In a footnote to the list of known Mersenne prime numbers which can be found here, we read that the "ranking" therein is a provisional one since not all possible exponents between $37 \, 156 \, 667$ ...
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Are fully general Frobenioids necessary?

Mochizuki's notion of a Frobenioid introduced in The geometry of Frobenioids I is rather elaborate. However, he also introduces a myriad of further properties that a Frobenioid may satisfy, and his ...
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Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)

Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?
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The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206). If ...
Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that Each coefficient is bound in absolute value by $B$ Degree of each variable in any monomial is bound by $d$ Total degree is $d'$ $f(x_1,\... 4answers 657 views Different derivations of the value of$\prod_{0\leq j<k<n}(\eta^k-\eta^j)$Let$\eta=e^{\frac{2\pi i}n}$, an$n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ... 1answer 210 views On the “infinitely often in” relation between subsets of$\mathbb{N}$Let${\mathbb N}$denote the set of positive integers, let$A,B\subseteq \mathbb{N}$. For$n\in\mathbb{N}$we set$n+A:=\{n+a: a\in A\}$. We say that$A$is infinitely often in$B$if the set$$\big\{... 1answer 454 views A question about Galois representations Let$K$be a number field and$(\rho,V)$,$(\rho',V')$be two Galois representations of$\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer$n$we have$\mathrm{Sym}^n\rho\...
I'm interested in locating dimension formulae for (more general) Jacobi forms associated with a lattice $L$ (where the Jacobi forms of Eichler-Zagier correspond to $L=A_1$). Unfortunately, the ...