All Questions
7 questions
13
votes
1
answer
442
views
Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?
Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...
- 41.9k
9
votes
1
answer
698
views
Hensel's lemma, Bezout's identity, and the integers
Factorization in the ring $\mathbb{Z}[x]/(x^2+1)\mathbb{Z}[x]\cong \mathbb{Z}[i]$ is well known. For instance, $5$ and $13$ (and any prime $\equiv 1\pmod{4}$) are no longer prime.
The factorization ...
- 18.7k
8
votes
6
answers
2k
views
How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?
Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
- 1,703
7
votes
1
answer
1k
views
Expressing primes $p\equiv 1 \pmod 3$ in the form $p = x^2 + xy + y^2$
Fermat famously showed that the only primes $p$ of the form $x^2 + y^2$ are the primes such that $p \equiv 1 \mod{4}$. Furthermore, we now know “effective” versions of Fermat's theorem, i.e. given a ...
- 1,703
7
votes
2
answers
606
views
convergence in $\hat{\mathbb{Z}}$, modulo prime power
The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16).
Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$
by $a_0=b, a_{n+1}=2^{a_n}$. Prove that $...
- 101
3
votes
2
answers
549
views
Primary structures in $\mathbb Q$
I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
- 7,500
2
votes
0
answers
182
views
Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
- 29