# 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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### Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
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### Most “natural” proof of the existence of Hilbert class fields

Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...
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### Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation $$[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=\dots[m], \qquad m\in\mathbb Z_{>0}.$$ Consider the following trigonometric numbers:...
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### Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...
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### Is every positive integer the rank of an elliptic curve over some number field?

For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$? It's easy to show that the set of such $n$ is unbounded. But can one show that ...
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### Rationality of sum of reciprocals of irreducible polynomial

(Sorry for my poor english..) I have a question in number theory. (Just my curiosity) Let $f(z)\in \mathbb{Z}[x]$ be an irreducible polynomial with degree is larger than or equal to 2. Then is the ...
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### Finiteness of etale cohomology for arithmetic schemes

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
907 views

### Can a number be palindromic in more than 3 consecutive number bases?

$2017:$ Was initially asked on MSE - but wasn't solved or updated there since. Update $2019$: I've returned to this problem, made some progress and updated the post here. (I've basically rewritten ...
The "quantum" version qTSPP of the number of totally symmetric plane partitions, contained in the cube $[0,n]^3$, is enumerated by f_n(q):=\prod_{j=1}^n\prod_{k=1}^j\prod_{\ell=1}^k\frac{1-q^{j+k+\...
### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space
Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...