# 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

11,043
questions

**-5**

votes

**0**answers

55 views

### equation x+(2x+3)h=k [on hold]

you know k. k,h are natural, x is integer ≥0, how i can know x?
example if k=23, x=2
This equation comes from the fact that every prime number can be constructed from 2k + 3, but there are k that do ...

**7**

votes

**3**answers

173 views

### Number of matrices with bounded products of rows and columns

Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not ...

**0**

votes

**0**answers

20 views

### Equations involving quasiperfect numbers: a first search of odd solutions for this type of equations or well succinct reasonings about these

In this post we study the following equations that involve quasiperfect numbers, denoted as $x$, that are integers such that the sum of all its positive divisors is equals to $2x+1$, and certain ...

**6**

votes

**0**answers

65 views

### On the first sequence without collinear triple

Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...

**-4**

votes

**0**answers

29 views

### PISANO PERIOD and its application and calculation [on hold]

WHAT IS PISANO PERIOD AND ALSO HOW TO CALCULATE PISANO PERIOD?
WHAT IS IT RELATION WITH FIBONACCI SERIES?
I ENCOUNTERED THIS TERM DURING A CODING ASSIGNMENT OF A FIBONACCI MOD ANOTHER NUMBER.

**15**

votes

**1**answer

307 views

### The space of cusp forms for $\mathrm{GL}_2$ over ${\mathbf{F}}_q(T)$

This question is about automorphic forms for the group $\mathrm{GL}_2$, over a rational function field. Let's say $\mathbf{F}_q$ is a finite field, and $X=\mathbf{P}^1_{\mathbf{F}_q}$ is the ...

**2**

votes

**0**answers

78 views

### Is this subset of a rigid space an admissible open?

Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.
Consider the sets $U := \{ |T_1| < 1\}$ and $V :...

**-4**

votes

**1**answer

184 views

### Numbers representable as in the famous IMO question number 6 (1988)

The famous problem number 6 of the 1988 International Mathematical Olympiad is about showing that if $a,b$ are non-negative integers such that $\frac{a^2+b^2}{ab+1}$ is an integer, then it is a square ...

**2**

votes

**1**answer

129 views

### An asymptotic formula for a sum involving powers of floor functions

Let $\theta \geq 0$ and consider the sum $$\sum_{n \leq x} \left\lfloor \frac{x}{n} \right\rfloor^{-\theta}.$$.
I have seen the claim that there is a constant $c(\theta)$ (depending on $\theta$!) ...

**0**

votes

**0**answers

33 views

### Squares in $\prod_{m=1}^{M}\left(1+(\Phi_n(m))^2\right)$, where $\Phi_n(x)$ is the the $n$th cyclotomic polynomial, and a related sequence

In this post we denote the $n$th cyclotomic polynomial as $\Phi_n(x)$, as reference we've for example the Wikipedia Cyclotomic polynomial. And for each integer $m>1$ we denote the product of its ...

**6**

votes

**1**answer

148 views

### Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators

I am reading Langlands' pape Euler Products and have a few questions. Let $G$ be a split adjoint semisimple group over $\mathbb Q$. If $p$ is a place of $\mathbb Q$, finite or infinite, let $G_{\...

**19**

votes

**0**answers

424 views

### On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...

**0**

votes

**0**answers

76 views

### On the number of solutions of $\gcd\left({2n\brace n},105\right)=1$, over integers $n\geq 1$

In this post we denote the Stirling numbers of the second kind as ${n\brace k}$. I present a variant of the problem showed in the penultimate paragraph of section B33 of [1] (see also the cited ...

**1**

vote

**1**answer

80 views

### Chinese Remainder Theorem for Remainder Intervals

Given $n$ natural numbers $m_1,\dots,m_n$ and $n$ remainder intervals $[a_1,b_1],\dots,[a_n,b_n]$ holding $a_i < b_i$ for all $i\leq n$ the task is to search for the smallest natural number $x$ ...

**3**

votes

**1**answer

238 views

### Is the diophantine equation $3x^2+1=py^2$ always solvable for each prime $p\equiv 13\pmod{24}$?

In Question 337879, I conjectured that for any prime $p\equiv3\pmod4$ the equation $$3x^2+4\left(\frac p3\right)=py^2\tag{1}$$
always has integer solutions, where $(\frac p3)$ is the Legendre symbol. ...