All Questions
13 questions
0
votes
0
answers
196
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Sum of squares squared in an arithmetic progression
Let $r(n)$ be the number of ways to write $n$ as a sum of two squares and $(a,q)=1$.
What is known about
$$
\sum_{n \le x,n \equiv a (\text{mod} \, q)} r(n)^2 \quad?
$$
I am looking for uniform ...
- 130
19
votes
0
answers
523
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
- 109k
9
votes
1
answer
1k
views
Sums of two squares in arithmetic progressions
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
- 143
13
votes
1
answer
2k
views
For a proof of the three-square theorem without using Dirichlet's theorem on primes in arithmetic progressions
The three-square theorem states that $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three squares if and only if it is not of the form $4^k(8m+7)$ ($k,m\in\mathbb N$). This was first proved by ...
- 15.6k
14
votes
0
answers
481
views
If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?
Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
- 109k
1
vote
1
answer
504
views
Fermat two square and Lagrange four square via Hardy-Littlewood circle method [closed]
Fermat two square: An odd prime p is expressible as
${\displaystyle p=x^{2}+y^{2},\,}$
with $x, y$ integers, if and only if
${\displaystyle p\equiv 1{\pmod {4}}.}$
Lagrange four square: Every ...
- 841
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...
- 3,062
12
votes
2
answers
499
views
"Pythagoras number" for integral matrices
It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $...
- 3,031
8
votes
2
answers
675
views
The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$
I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
user40023
3
votes
0
answers
338
views
The hypertriangular function of $n$
I'm looking for papers or recent results on the hypertriangular function of $n$:
$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$
This is A001923 in the OEIS.
I don't have much experience with ...
- 181
6
votes
1
answer
1k
views
Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$
The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it:
Problem 19.5 (p. 224) ...
- 5,191
19
votes
1
answer
2k
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Legendre and sums of three squares
The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed
to have given a proof ...
- 32.5k
9
votes
1
answer
2k
views
Sums of two squares in (certain) integral domains
While giving the first of eight lectures on introductory model theory and its applications yesterday, I stated Hilbert's 17th problem (or rather, Artin's Theorem): if $f \in \mathbb{R}[t_1,\ldots,t_n]$...
- 65.4k