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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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      -5
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      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 ...
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      votes
      3answers
      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 ...
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      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 ...
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      votes
      0answers
      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. ...
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      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
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      1answer
      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 ...
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      votes
      0answers
      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
      1answer
      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
      1answer
      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 $\theta$!) ...
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      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
      1answer
      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
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      0answers
      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
      0answers
      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
      1answer
      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
      1answer
      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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