43 votes
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Lagrange four squares theorem

The set $X$ doesn't have to be the set of non-negative integers. This was known already to Härtter and Zöllner in 1977, who constructed an $X$ of the form $\{ 0, 1, 2, \ldots \} \setminus S $ for an ...
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27 votes
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Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?

Yes, it is true that $\{ x^{4} + y^{2} + z^{2} : x, y, z \in \mathbb{Z}[i] \} = \{ a + 2bi : a, b \in \mathbb{Z} \}$. Indeed, one can even take $x$ to be either $0$ or $1$ in all cases. Because $y^{2}+...
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  • 17.8k
26 votes
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Distribution of $a^2+\alpha b^2$

There is a positive proportion of such integers and this follows from work on the Berry-Tabor conjecture due to Eskin, Margulis and Mozes (there is work by many others on this; see these papers for ...
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  • 42.4k
26 votes
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Primes that are sums of two squares with constraints on the squares

Hecke showed that there are infinitely many primes of the form $p=a^2+b^2$ with $a = o(\sqrt{p})$. Ankeny improved this to $o(\log p)$, conditional on the Extended Riemann Hypothesis. Harman and Lewis ...
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20 votes
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SOS polynomials with rational coefficients

In general you can't write $p = f^2 + g^2$ in ${\bf Q}[x]$ at all, let alone do so efficiently. For example, $2 x^2 + 3$ is positive for all $x$ (and is the sum of three squares, $(x+1)^2 + (x-1)^2 + ...
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19 votes
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Many representations as a sum of three squares

The formula for $r_3(n)$ essentially connects this with a class number of an imaginary quadratic field, or (apart from the $\sqrt{n}$ scaling) with the value of an $L$-function at $1$. So your ...
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  • 42.4k
19 votes
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Lagrange four-squares theorem --- deterministic complexity

As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a ...
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  • 28.7k
18 votes

Many representations as a sum of three squares

Let me restrict to the number of primitive representations $$r_3^\ast(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n\ \text{and}\ \gcd(a,b,c)=1 \}\right|.$$ Note that $r_3(n)$ can be easily ...
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  • 85.6k
18 votes

Sum of squares and divisibility

This is not a complete answer, but just a way to transform the problem into one that can be attacked by brute force in some known way. Write $d_i^2=N/n_i$. Then your relation becomes $$\frac{1}{n_1}+ \...
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17 votes
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Realization of numbers as a sum of three squares via right-angled tetrahedra

These numbers are not very prevalent and the ratio in question goes to zero. Note first that by Legendre's theorem, a positive proportion of the numbers below $n$ may be expressed as a sum of three ...
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  • 42.4k
17 votes
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Sums of two squares in arithmetic progressions

The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write $$\sum _{n\...
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16 votes

Jacobi's theorem on sums of two squares (reference request)

Using class field theory one can prove for every integer $n>0$ there exists a monic polynomial $f$ of degree $h(-4n)$* such that for any odd integer $m$ coprime to $n$ we have the following ...
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  • 2,371
16 votes

Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

Answer to Question 1. Yes, $E(\mathbb{Z})=F$. The inclusion $F\subseteq E(\mathbb{Z})$ follows from $E\subseteq E(\mathbb{Z})$ and Dickson's result $E=F$. It remains to show $E(\mathbb{Z})\subseteq F$...
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15 votes

$x^2+7y^2=2^n$ and sums of four squares

The Diophantine equation $x^2 + 7y^2 = 2^n$ is "trivial" in the sense that its solutions can be described in a very simple way (see boxed formula near the end for solutions in odd numbers ...
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  • 41.9k
15 votes
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Representing $x^3-2$ as a sum of two squares

The answer is similar to one provided here. The approach is elementary and proves stronger fact that it can be expressed as sum of two coprime squares. We consider the product $(n+2)(n^3-2)$ which is ...
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14 votes

A generalization of partition function to the sums of squares

This asymptotic was stated by Hardy and Ramanujan, but without proof. The first proof of this asymptotic was given by Wright in 1934 [1], by a rather complicated argument. A much simpler approach ...
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  • 5,794
13 votes

For a proof of the three-square theorem without using Dirichlet's theorem on primes in arithmetic progressions

I am not sure if Legendre's proof was complete (see this related MO entry), but surely Gauss gave a proof without Dirichlet's theorem (which came decades later). A nice transparent proof (without ...
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  • 85.6k
10 votes
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The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$

Yes, this is standard. More generally, we have the following Theorem. Let $p$ be an odd prime, and let $a_1,\dots,a_k\in\mathbb{F}_p^\times$. Then the number of solutions of the equation $a_1x_1^2+\...
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  • 85.6k
9 votes

Why sum of three squares of real polynomials is a sum of two squares?

Theorem 4.2 from this paper says that in general you will need the coefficients of the two squares to be algebraic numbers of exponential, in $\deg (f^2+g^2+1)$, degrees. More precisely, if the Galois ...
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9 votes

Lagrange four squares theorem

I know this question has already been sufficiently answered, but I would like to mention an explicit construction of Choi-Erdős-Nathanson, as opposed to the probabilistic proofs that are common in ...
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  • 1,474
9 votes

Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

given your interest: the list of all $A x^2 + B y^2 + C z^2$ with ordered positive coefficients, such that the represented numbers can be described by congruences Ummm. Allowing mixed terms, all 913 (...
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  • 24.5k
8 votes

Jacobi's theorem on sums of two squares (reference request)

The requested generalization of Jacobi's two-square theorem is a remarkably recent result: N. Bagis and M.L Glasser, On the Number of Representations of Integers by various Quadratic and Higher Forms (...
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8 votes
Accepted

Sum of squares and divisibility

I hope so. But please double check (or, better, simplify) the argument below. Denote $N=qs^2$ for $q$ squarefree. Then each $d_i$ divides $s$, say $d_i=s/m_i$ and we get $$q=1/s^2+\sum_{i=1}^r 1/m_i^2,...
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  • 88.6k
8 votes
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Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?

Everything becomes simpler if add some parameters and start the sum at $k=-n$ instead of $k=0$. Note that if $k$ is a negative integer with $-n\le k \le -1$ then $L_n^k(x)$ is a polynomial in $x$ ...
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  • 13.3k
7 votes
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How many finite subsets in $\mathbb{Z}^d$ have a given sum of squares?

$|\varphi^{-1}(m)|$ is a coefficient at $x^m$ in the product $$ \prod_{s\in \mathbb{Z}^d} (1+x^{|s|^2}):=F(x) $$ Hence for $0<x<1$ we have $$|\varphi^{-1}(m)|\leq x^{-m} F(x).$$ Minimising RHS ...
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  • 88.6k
7 votes
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How often is the value of a quadratic polynomial equal to a sum of two integer squares?

This problem has been addressed in a paper of Friedlander and Iwaniec in Acta Mathematica 1978, called Quadratic polynomials and quadratic forms. Under general conditions they count the number of ...
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  • 42.4k
6 votes

The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$

I just want to share a character-free proof - very accessible and too long for a comment (also - probably not original). Of course, it's just a restatement of a character-full proof. Let $\vec{v} \in ...
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6 votes
Accepted

"Pythagoras number" for integral matrices

Here is an answer (see the last point). It differs from what I had been claiming in my first post. There I was saying that any positive bilinear module $\Lambda$ over $\mathbf Z[\frac 12]$ was ...
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  • 1,960
6 votes

Specific quaternary quartic that is positive semi-definite but not sum of squares

Yes, there is such an example in [Choi, Positive semidefinite biquadratic forms, §4]. It is proven there that the polynomial $$x^4+y^4-2(x^2+y^2)zw+z^2w^2+2(x^2z^2+y^2w^2)$$ is positive semidefinite, ...
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6 votes
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Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

This Wikipedia page has many references: https://en.wikipedia.org/wiki/Positive_polynomial, including for example Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and ...
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