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 $\{...

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}{(\...

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 ...

10
votes

### Accelerating convergence for some double sums

This is easy to do with PARI/GP.
Here is my code
...

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 ...

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 ...

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\...

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\...

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 "...

4
votes

Accepted

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 ...

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.

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.

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 | ...

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 ...

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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