15 votes

Polynomial values are powers of two

I'll prove a stronger statement. Let $S$ be a finite set of primes. I claim there is a $c_{n,S}$ such that a polynomial $f$ with rational coefficients cannot take only values that are $S$-units on $\{...
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13 votes

Accelerating convergence for some double sums

With Mathematica I can first sum the series over $\ell$ to get a closed-form expression in terms of polygamma functions, $$Z(2,2)= \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^2} \frac{2\ell+3}{(\...
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13 votes

Polynomial values are powers of two

Yes, such $c_n$ is bounded by something effective. Below is a cubic bound, which probably may be improved. (Update: see $n^2\log n$ upper bound by Will in the comments.) Assume that $f(x)$ is a power ...
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10 votes

Accelerating convergence for some double sums

This is easy to do with PARI/GP. Here is my code ...
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  • 1,923
9 votes

Accelerating convergence for some double sums

Here is a little Maple. Note that using "sum" on the inside causes it to find an algebraic expression for the sum over $\ell$ and using "Sum" on the outside tells it to not try to ...
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8 votes

On Cramér's theorem about roots of Zeta function

Q1: There is a functional relation for $V(z)$, but no "explicit expression" I know of. Q2: Cramér's paper is from 1919, a modern and more extensive treatment is in On Cramér's theorem for ...
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7 votes

Surprisingly long closed form for simple series

Let $a:=A$. For $a\ge2$ and $r\in\{0,\dots,a-1\}$, we have $$s(r):=\sum_{n=1}^\infty\frac1{a^n}\frac1{an+r} =\sum_{n=1}^\infty\frac1{a^n}\int_0^1 dx\, x^{an+r-1} \\ =\int_0^1 dx\, \sum_{n=1}^\infty\...
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6 votes

Surprisingly long closed form for simple series

The intuition becomes clear when we evaluate the series by bare hands. For $r\in\{1,\dotsc,A\}$ we have \begin{align*}\sum_{n=0}^\infty \frac{1}{A^n(An+r)}&= \sum_{\substack{m\geq 1\\m\equiv r\...
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5 votes

Primality test for $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ and $\frac{(10 \cdot 2^n)^m+1}{10 \cdot 2^n+1} - 2$

It's often the case with such tests that the "only if" part is more or less easy to prove, while the "if" part is inaccessible for proving or disproving. Below I prove the "...
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4 votes
Accepted

Has any one seen this sum of roots of unity before?

Using the sagemath code, ...
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  • 56
3 votes

Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to the conjecture of Schinzel and Tijdeman (Schinzel, Α., Tijdeman, R., On the equation $y^n = P(x)$, Acta Arith. 31 (1976), 199-204), if a polynomial $P(x)$ with rational coefficients has ...
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3 votes

Primality test for numbers of the form $\frac{a^p-1}{a-1}$?

The test fails already for $p=3$ and $a=7$, claiming that $57=3\cdot 19$ is prime.
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2 votes

Surprisingly long closed form for simple series

$$f(A)=\frac 1{A^2}\left(\Phi \left(\frac{1}{A},1,1+\frac{1}{A}\right)-\Phi \left(\frac{1}{A},1,2-\frac{1}{A}\right)\right)$$ where $\Phi$ is Lerch function.
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2 votes
Accepted

A condition on $(a_{j})_{j\in \mathbb{N}}$ so that for all $x \in \mathbb{R}$ we have $\min_{1 \leq j \leq N}\|a_{j}x\|=o(1)$

I will expand upon @mathworker21's comment. Let $\theta = \frac{1 + \sqrt{5}}{2}$, and define the sequence $a_d$ greedily to be the smallest positive integer greater than $a_{d - 1}$ such that $d | ...
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  • 2,027
1 vote

Diophantine equations with arithmetical functions

This is in response to your question as to whether this is an "interesting or good topic of research." There is no answer to such a question. If you find it interesting, that makes it ...
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1 vote

Diophantine equations

Pythagorean triples can be used to construct right angles and so had applications to surveying in the ancient world, see this link.
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