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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

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        1answer
        59 views

        Maps to additive group scheme

        Let $\underline{\mathbb{Q}_p/\mathbb{Z}_p}$ be constant p-divisible group over $\mathbb{F}_p$. And let $\mathbb{G}_a$ be the additive group over $\mathbb{F}_p$. Let me prove $$ Hom(\underline{\mathbb{...
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        41 views

        Primes in this region

        Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$ Using Chinese remainder theorem we can show that : $$\#\{(...
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        1answer
        85 views

        Prime divisors of $\prod_{i=1}^n (i^2+1)$

        Is it true that for every positive integer $n$ there is a prime $p>n,$ which divides $\prod_{i=1}^n (i^2+1)$ ?
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        114 views

        $|L'(1,\chi)/L(1,\chi)|$

        Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$? Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\...
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        64 views

        Is it possible to stab every permutation of any four element subset of $D_n$ with less than $n/2$ elements?

        Say for a permutation group $G$ over $n$ that a set $S\subset \{1,\ldots,n\}$ is G-stabbed by $X\subset \{1,\ldots,n\}$ if for every $g\in G$ we have $gS\cap X\ne \emptyset$. Is there for every $|S|...
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        1answer
        143 views

        Power series equation with solution $1/e$ [on hold]

        As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root. Are there classical equations of the form $$\sum_{i=0}^{\infty} a_ix^i =1$$ that have $e$ ...
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        74 views

        An integral involving the Chebyshev $\psi$ function

        Consider the Chebyshev $\psi$ function $\psi(x)=\sum_{p^r \leq x} \log p$ on prime powers. Define $I=\int_{1}^{\infty} (\psi(x)-x)x^{-2} \mathrm{d}x$. My question is, does I exist ? Someone has ...
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        44 views

        Half-integral weight slash operator

        I am aware that the correct way to look at modular forms of half-integral weight is on the metaplectic cover of $\mathrm{SL}_2(\mathbb R)$. Assume however that we insist on considering them as ...
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        2answers
        340 views

        Witt vectors addition confusing

        I raise this confusing because I try to understand the witt vectors for characteristic not equal to p. Let us assume p=2. The Witt Polynomials is explicitly given by $$ S_0=X_0+Y_0 $$ $$ S_1=X_1+...
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        Is it possible to stab (every rotation of) any four element subset of $\mathbb Z_n$ with less than $n/2$ elements?

        Say that $S\subset \mathbb Z_n$ is stabbed by $X\subset \mathbb Z_n$ if for every $t$ we have $(S+t)\cap X\ne \emptyset$. Is there for every $|S|=4$ an $|X|<n/2$ that stabs it? My motivation ...
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        0answers
        111 views

        Kronecker limit formula, modular curves, and the class number problem

        Let $$Q(x,y)=ax^2+bxy+cy^2$$ be a positive definite quadratic form with $a>0$ and $D=b^2-4ac<0$. Let $$\zeta_Q(s)=\sideset{}{'}\sum_{m,n}Q(m,n)^{-s},$$ the accent indicating that $(0,0)$ is ...
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        2answers
        245 views

        Solutions of $y^2=\binom{x}{0}+2\binom{x}{1}+4\binom{x}{2}+8\binom{x}{3}$ for positive integers $x$ and $y$

        I was interested in create and solve a Diophantine equation similar than was proposed in the section D3 of [1]. I would like to know what theorems or techniques can be applied to prove or refute that ...
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        63 views

        compute class number of real quadratic field without first computing fundamental unit

        Let K be the quadratic field Q(d) with d > 0. Let u be the fundamental unit, h the class number, R the regulator, which by definition is abs(log(abs(u))). The only way I know to compute u is via the ...
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        Level Lowering Galois representations over Totally real fields

        Let $F$ be a totally real number field and $\mathbb{F}$ a finite field. Let $\bar{\rho}:\text{Gal}(\bar{F}/F)\rightarrow \text{GL}_2(\mathbb{F})$ an irreducible Galois representation arising from a ...
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        0answers
        125 views

        Some more super-congruences involving roots of unity

        Let $K$ be a number field and $c_1,…,c_r \in K$ and $x_1,…,x_r\in K$ some fixed coefficients. Let $S$ be the infinite set of all rational primes, which split completely in $K\mid\mathbb{Q}$. Suppose ...

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