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 ...
Ofir Gorodetsky's user avatar
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}+...
Jeremy Rouse's user avatar
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 + ...
Noam D. Elkies's user avatar
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 ...
Tony Huynh's user avatar
  • 31.5k
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}+ \...
Francesco Polizzi's user avatar
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\...
Ofir Gorodetsky's user avatar
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 ...
Ethan Splaver's user avatar
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 $E(\mathbb{Z})\subseteq F$...
Philipp Lampe's user avatar
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 ...
Hhhhhhhhhhh's user avatar
  • 1,032
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 ...
KConrad's user avatar
  • 49.6k
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 ...
Thomas Bloom's user avatar
  • 6,608
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 ...
GH from MO's user avatar
  • 99.2k
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 ...
Henri Cohen's user avatar
  • 11.8k
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 ...
GH from MO's user avatar
  • 99.2k
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 (...
Will Jagy's user avatar
  • 25.4k
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 ...
Woett's user avatar
  • 1,533
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 ...
Dima Pasechnik's user avatar
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,...
Fedor Petrov's user avatar
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}\}| =...
Ofir Gorodetsky's user avatar
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 (...
Carlo Beenakker's user avatar
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 ...
Lucia's user avatar
  • 43.3k
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 $x$ ...
Ira Gessel's user avatar
  • 16.2k
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 ...
Zach Teitler's user avatar
  • 6,197
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 ]...
Pavel Kozlov's user avatar
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,⋯,\...
Bazin's user avatar
  • 15.2k
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 $...
Mark Lewko's user avatar
  • 11.8k
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, ...
Olivier Benoist's user avatar
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,...
Josiah Park's user avatar
  • 3,177
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 ...
Lucia's user avatar
  • 43.3k
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 ...
Geoff Robinson's user avatar

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