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        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

        4
        votes
        1answer
        194 views

        Upper bound for the number of solutions of a Diophantine equation

        Consider the Diophantine equation $$k^2 + k - \sigma (\ell^2 + \ell) = m,$$ where $N \leq k \leq 2 N$, $L \leq \ell \leq 2 L$, $m \in \mathbb{Z}$ and $\sigma \in \mathbb{R}$. For which values ...
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        votes
        0answers
        67 views

        A Diophatine equation question [on hold]

        The question is the following. Consider the Diophantine equation $3^x + 2y = 5^z$. Then $x, y , z$ must necessarily be even numbers ?
        3
        votes
        1answer
        140 views

        Why the level of a half integral weight modular form must be a multiple of 4?

        Let $\Gamma_0(N)$ be the Hecke congruence subgroup. Let $S_{k+1/2}(\Gamma_0(N))$ be the space of holomorphic forms of weight $k+1/2$ on $\Gamma_0(N)$, where $k\in\mathbb{N}$. How to prove that if $S_{...
        4
        votes
        1answer
        112 views

        The exceptional eigenvalues and Weyl's law in level aspect

        The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...
        11
        votes
        1answer
        158 views

        Most points on a degree $p$ hypersurface?

        Let $p$ be a prime. Let $f \in \mathbb{F}_p[x_1, \ldots, x_n]$ be a homogenous polynomial of degree $p$. Can $f$ have more than $(1-p^{-1}+p^{-2}) p^n$ zeroes in $\mathbb{F}_p^n$? Basic observations: ...
        1
        vote
        0answers
        110 views

        Level structures in deformation spaces of $p$-divisible groups

        I am reading (parts of) the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein, and I am stuck at understanding the definition of level structures in Rapoport-Zink spaces (cf. Definition ...
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        votes
        1answer
        56 views

        Primes to base 30: n*30+{1,7,11,13,17,19,23} or n*30+{7,11,13,17,19,23,29} not possible to be prime for all values in {}, why? [on hold]

        Using base 30 I just checked it for the first 2^32 numbers and it seems there is no such sequence that n*30 + 1, n*30+7, n*30+11, n*30+13, n*30+17, n*30+19, n*30+23 are all primes for some n. Another ...
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        votes
        0answers
        59 views

        References for the extension of Euler's phi function to number rings

        Can anyone post a self-contained reference concerning the extension of the Euler phi function to number rings and its basic properties (reminiscent of those that the classic Euler phi function has)? ...
        1
        vote
        1answer
        124 views

        A Wolstenholme type congruence

        Consider the following congruence: For $p\geq 5$ prime and every $n,\nu\in\mathbb{N}$ we have \begin{align*} 0\equiv\sum_{k=1\atop p\nmid k}^{pn-1}\frac1k \binom{pn(\nu+1)-k-1}{pn\nu-1} \mod p^{2(\...
        5
        votes
        0answers
        89 views

        Extension of Erdos-Selfridge Theorem

        Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem $\text{Theorem 1:}$ The product of two or more consecutive positive integers is ...
        10
        votes
        1answer
        734 views

        Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

        I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following: Lemma: if L/K is an abelian ...
        2
        votes
        0answers
        92 views

        Behavior of L-function in families of elliptic curves

        I'm curious about how the L-functions of elliptic curves behave as the elliptic curves vary in families. In other words, if we regard the L-function $L(E_{/K},s)$ of an elliptic curve $E$ over a ...
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        vote
        0answers
        52 views

        Algorithm for testing Fermat number divisors [migrated]

        The other day, I noticed a pattern within the factors of Fermat numbers from which I developed an algorithm that can quickly determine whether a number 'n' divides any Fermat number (not a specific ...
        20
        votes
        2answers
        2k views

        Intuition behind counterexample of Euler's sum of powers conjecture

        I was stunned when I first saw the article Counterexample to Euler's conjecture on sums of like powers by L. J. Lander and T. R. Parkin:. How was it possible in 1966 to go through the sheer ...
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        votes
        0answers
        172 views

        Can we write each integer $n>12$ as $x+y+z$ with $T_x+T_y+T_z$ a triangular number?

        Recall that the triangular numbers are those natural numbers $$T_n:=\frac{n(n+1)}2\quad \ (n\in\mathbb N=\{0,1,2,\ldots\}).$$ It is well known that each $n\in\mathbb N$ can be written as the sum of ...

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