# 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

10,465 questions
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### Divisor bound for $r_2$ off the origin

If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
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### Does there exist $a_0$, such that $\{a_n\}_{n=0}^{\infty}$ is unbounded?

Suppose $\{a_n\}_{n=0}^{\infty}$ is a sequence, defined by the recurrence relation $$a_{n+1} = \phi(a_n) + \sigma(a_n) - a_n,$$ where $\sigma$ denotes the divisor sum function and $\phi$ is Euler'...
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### Understanding an equality in the paper “Class groups, totally positive units, and squares”

This is an excerpt from the paper "Class groups, totally positive units, and squares" (page 36). I am struggling to understand the last equality $|K^{(1)}_{2}:K|=|\overline{O}_K^{+}|$, the bar ...
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### Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates

Let $S$ be the set of positive integers of the form $2^a3^b 5^c 7^d$. I need information about the cardinality of the intersection of $S$ and its translates. In particular, is $S \cap (S+t)$ ...
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### Valuation of congruent elements in a local division ring

Let $K$ be a complete local division ring (note $v$ its valuation). For $x,y\in K$ ($y\ne0$), one puts $x^y=yxy^{-1}$. Let $r\in\mathbb N$. Consider $x,y\in K$ and $a,b\in K^*$ such that $v(x-y)\ge r$ ...
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### Largest cardinality $n$ of a subset $A$ of $\{1,2,\ldots,M\}$ such that $(A+A) \cap A$ is empty

Assume that the set $A$ does not have simple structures (such as the case that when all elements are odd numbers in $[1,M/2]$ then all sums are even thus there are no solutions, as pointed out by @...
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### Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$?

There may be some technical issues with the question, but hopefully what I mean is clear... Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0) ...
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### Hensel lemma and rational points in complete noetherian local ring

Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal. If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
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### Intuitive reason why the $j$-invariant is a cube?
Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let \$ \gamma_2(\tau)=j(\tau)^{...