43
votes

Accepted

### 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 ...

27
votes

Accepted

### 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}+...

20
votes

Accepted

### 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 + ...

19
votes

Accepted

### 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 ...

19
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}+ \...

17
votes

Accepted

### 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\...

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 ...

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 $E(\mathbb{Z})\subseteq F$...

16
votes

Accepted

### 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 ...

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 ...

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 ...

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 ...

11
votes

### Is there an upper bound on the number of representations as a sum of squares?

It is known since Gauss that
$$r_3(n)=12\dfrac{h(D)}{w(D)/2}(1-(D/2))\sum_{d\mid f}\mu(d)(D/d)\sigma(f/d)\;,$$
where $-n=D(2^vf)^2$, $D$ a fundamental discriminant, $v\ge-1$, $h(D)$ is the class ...

11
votes

Accepted

### Jacobi symbols for two-square sums of primes

The first observation follows from the law of quadratic reciprocity. Indeed, assume that $p\equiv 1\pmod{8}$ and $p=A^2+B^2$. Let $A'$ denote the odd part of $A$. Then $p\equiv B^2\pmod{A'}$, and ...

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 (...

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 ...

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 ...

9
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,...

9
votes

Accepted

### A simple way to bound the density of sums of two odd squares

A number $n$ is a sum of two squares if and only if $2n$ is (this goes back to Fermat and Euler). This implies
$$S_{\text{even}}(x):=|\{n^2+m^2: n,m \ge 0, \, n^2+m^2 \le x,\, n^2+m^2 \text{ even}\}| =...

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 (...

8
votes

Accepted

### 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 ...

8
votes

Accepted

### 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 $x$ ...

7
votes

Accepted

### 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 ...

7
votes

Accepted

### Efficient computation of $\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor$

Following by comment of Alexey Kulikov we could split our sum in the next way:
$$\sum_{i=1}^{[\sqrt{n}]} i^2\left [\frac{n}{i^2}\right ]=
\sum_{[n/i^2]>[\sqrt[3]{n}]} i^2\left [\frac{n}{i^2}\right ]...

7
votes

### Symmetric version of Hilbert's seventeenth problem?

There is paper by Georges Glaeser [MR0143058], published by the Annals of Mathematics in 1963, in which he proves that
Every symmetric smooth fonction on $\mathbb R^n$ is equal to
$g(\sigma_1,⋯,\...

6
votes

### Fermat two square and Lagrange four square via Hardy-Littlewood circle method

Let me sketch a proof using Fourier analysis that I quite like, although perhaps not exactly what I would call the Hardy-Littlewood circle method.
Let $r(n)$ denote the number of representations of $...

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, ...

6
votes

### Sums of squares of primes

The sequence $f(n)$ given by those smallest integers which can be written in $n$ ways as the sum of squares of three primes has first eleven values,
$$12,219,363,699,1179,2019,2259,3891,4059,6459,...

6
votes

### Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$

Problems like Question 1 go back to Euler and Gauss. This question is essentially asking which idoneal numbers are sums of two squares. There is a famous open problem going back to Euler and Gauss ...

6
votes

### Sum of squares and divisibility

The answer of Francesco Polizzi recasts the problem into a form in which known results prove at least that there are (at most) a finite number of exceptions.
For any positive integer $s$, E. Landau ...

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