# 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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### Good software for shortest vector for non-full rank lattices?

what is a good algorithm for shortest vector in non-full rank lattices in say dimensions $8$ to $64$? Is there a software package available?
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### Reference describing supersingular elliptic curves over algebraically closed field in characteristic 2

I'm looking for a reference for the fact that over an algebraically closed field of characteristic two, there is (essentially) only one supersingular elliptic curve. This fact appears on Wikipedia, ...
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### Extension of closed-form solvability of polynomial equations of x and exp to rational expressions?

Is my conjecture below true? It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page. It seems that Ferng-...
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### What are the asymptotics for $\sum_{r=0}^{x-1} \sum_{|\rho|\leq x} \frac{(x+r)^{\rho+1}}{\rho}$?

Let $f(x)=\sum_{r=0}^{x-1} \sum_{|\rho|\leq x} \frac{(x+r)^{\rho+1}}{\rho}$, where $\rho$ denotes a non-trivial zero of the Riemann zeta function. What are the upper and lower bounds for $f(x)$ ? ...
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### Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?

Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
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### Is new n-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$ ...
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### Radical Extensions - gcd and lcm [closed]

Given two natural numbers $m,n$ and say $\gcd(m,n)=g$, I would like to show that $$\Bbb Q(\xi _n )\cap \Bbb Q(\xi _m ) = \Bbb Q(\xi _g ).$$ One direction is trivial, and to show the other one I ...
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### Is there any relation between $h_{d_1},h_{d_2}$ and $h_{d_1d_2}$?

$h_{d_1}, h_{d_2}$ and $h_{d_1d_2}$ are class number of $\Bbb Q(\sqrt{d_1}),\Bbb Q(\sqrt{d_2}),\ and \ \Bbb Q(\sqrt{d_1d_2})$ respectively.
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### In how many ways to generate a vector with odd/even entries

Let $\bar v=(k_1, \ldots k_n)$ be a vector with entries $k_i$ from $\{0, \ldots, K\}$ with $K \in N$ and such that $\sum_{i=1}^n k_i=K.$ Find in how many ways we can generate vector $\bar v$ so that ...
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### Probability of factor of particular size

What is the probability that an integer $a$ picked uniformly in $[t/2,t]$ has a factor in $[t^{\alpha},2t^{\alpha}]$ where $\alpha\in(0,1)$? I am interested when $\alpha=1/3$ but if there is a general ...
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### Asymptotic for the probability that a number has $k$ prime factors less than $Q$

If we let $\omega_Q(n)$ denote the number of distinct prime factors of $n$ less than a bound $Q$, then what asymptotic formulas exist for $\Pr_{n\in\mathbb{N}}[\omega_Q(n)=k]$ as $Q\to\infty$ if $k$ ...
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### How smooth can this be?

If $a$ is an even integer then how smooth can $a^2-1$ be? Approximately how many integers in $a\in[0,t]$ are there such that $a^2-1$ is $k$-smooth?
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### Curious inversion formula in additive combinatorics

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions: $N_S(z)$ is asymptotic continuous version of the function counting the ...
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### Divisibility between values of polynomials

In this paper Corvaja and Zannier considered the following problem. Let $K$ be a number field, and let $S$ be a finite set of places of $K$ containing the infinite places and let $\mathcal{O}_S$ be ...
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### Primes which do not divide certain homogeneous polynomials

It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
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### Origin and variations of problem on $4xy-x-y$ being square

One of the forms in which the Diophantine equation in question can be found in the literature is this: Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
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### Density of integral points on affine cubic surfaces of a certain type

Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$. Is there a general ...
Is it true that for any even $k$ and $N,$ there always exist a Hecke eigenform with integer Fourier co-efficient of weight $k$ and level $N$ ?