All Questions
Tagged with prime-numbers quadratic-forms
13 questions
6
votes
2
answers
424
views
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
I already posted this question on MSE.
Using theorem $IV$ from this article, it is possible to prove that when $p$ is a prime such that $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the ...
- 387
6
votes
0
answers
381
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
6
votes
1
answer
1k
views
Set of quadratic forms that represents all primes
A SPECIFIC CASE:
Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$.
If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...
6
votes
0
answers
211
views
some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
- 841
5
votes
1
answer
208
views
applications of finding least quadratic nonresidue mod $p$?
I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$.
My question is that why it is so ...
- 841
5
votes
0
answers
229
views
Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
- 15.6k
3
votes
0
answers
111
views
On covering with Idoneal integers
$d\in\Bbb N$ is an idoneal integer if $N\in\Bbb N_{>1}$ can be written uniquely as $N=x^2\pm dy^2$ then $N=2^mp^n$ where $p$ is odd prime and $n\geq0$ and $m\geq0$ holds.
Let the $65$ known ...
user94040
18
votes
3
answers
5k
views
What is known about primes of the form $x^2-2y^2$?
David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
2
votes
0
answers
207
views
n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
- 1,639
2
votes
1
answer
107
views
Principally split primes with factors in arbitrarily small angular sectors
I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ ...
- 704
15
votes
2
answers
2k
views
Primes and $x^2+2y^2+4z^2$
A few months ago, I have asked a question about primes represented by ternary quadratic forms. I got two wonderful answers, which showed me how the theory was way richer and more complex that I ...
- 26k
20
votes
3
answers
2k
views
Primes of the form $x^2+ny^2+mz^2$ and congruences.
This is a sequel of this question where I asked for which positive integer $n$ the
set of primes of the former $x^2+ny^2$ was defined by congruences (a set of primes $P$ is defined by congruences if ...
- 26k
15
votes
1
answer
4k
views
The Green-Tao theorem and positive binary quadratic forms
Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a ...
- 25.7k