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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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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 ...
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 ...
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 ...
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. ...
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.
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 ...
78 views

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 ...
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  (see also the cited ...
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 $x$ ...
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. ...

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