# 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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### Can you solve the listed smallest open Diophantine equations?

In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
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### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
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### Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?

1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence? More formally, 2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA? Update (July 2010): So we have two ...
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### What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
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### Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.$$ Basic theorems about Dirichlet series ...
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### distance between powers of 2 and powers of 3

I expect this is a classical question, so feel free to point me to classical answers: what is the fastest-growing function $f(t)$ for which we know that $$|2^t - 3^{t'}| \ge f(\min(t,t')) \;?$$ In ...
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### Erik Westzynthius's cool upper bound argument: update?

Version 2 of this writeup is available, and includes a newer and simple upper bound thanks to MathOverflow 88777 as well as indirect references to future writeups. Details of further work ...
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### On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in ...
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