# Questions tagged [sums-of-squares]

The sums-of-squares tag has no usage guidance.

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### Is 100 the only Leyland number that is a square?

Leyland numbers (named for Paul Leyland) are positive integers of the form $x^y + y^x$ , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980
I thought ...

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**2**answers

207 views

### Sums of squares of primes [closed]

Question: What is the least number that is a sum of three squares of primes in exactly six ways?
... I know it is not research mathematics. Happy new year!
EDIT: Now that it is answered I should ...

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106 views

### Representation as $n=p^2+q^2-r^2$

What is known about the number of representations of a positive integer $n$ as
$$
\rho(n) = \# \{ (p,q,r): n=p^2+q^2-r^2\},
$$
where all the variables are primes?
What about the average number of ...

**2**

votes

**1**answer

108 views

### Sieve bound for the sum of two squares

Let $$S(n) = \sum_{p \le n} b(n-p),$$ where
$b(a)=1$ is $a$ is a sum of two squares of positive integers and $b(a)=0$ otherwise.
Trivially by PNT we have
$$S(n) \ll \sum_{p \le n}1 \le \frac{n}{\log n}...

**5**

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168 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 ...

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203 views

### Legendre's three-square theorem and squared norm of integer matrices

Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...

**11**

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366 views

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

Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...

**34**

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2k views

### Lagrange four squares theorem

Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that ...

**7**

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**1**answer

247 views

### Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...

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205 views

### Number of ways to write an integer as a sum of squares modulo $k$

Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, how many solutions are there in $\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ version ...

**9**

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**1**answer

305 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 ...

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457 views

### The number 1680 and Lagrange's four-square theorem

The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...

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39 views

### Points of intersection of summand of sums of squares of real polynomials

$\newcommand\R{\mathbb R}
\newcommand\Q{\mathbb Q}
$I am thinking of something related to Blekhermans 2012 paper Nonnegative Polynomials and Sums of Squares (Journal of the AMS, 25, 2012, 617-635).
...

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140 views

### An inequality involving sums of squares and sums of cubes

Assume that we have $m$ real numbers $x_1,x_2,...,x_m\in[0,1/4]$ satisfying the following equations:
$$ \sum_{i=1}^m x_i=a_1, \sum_{i=1}^mx_i^2=a_2,\sum_{i=1}^mx_i^3\geq a_3,$$
where $a_1,a_2,a_3$ are ...

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vote

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434 views

### Efficient sum of squares decomposition

Sum of 4 squares decomposition is the well-known result. I'm interested only in negative/non-negative separation with focus on efficiency and large numbers. I'm looking for alternatives or extensions ...

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428 views

### Does every integer $n>1$ have the form $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers?

Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: ...

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389 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?

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48 views

### For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials

Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following ...

**8**

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**1**answer

335 views

### Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime

Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number?
It seems that for every odd $m$ there are many $(a,b)\in \...

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150 views

### Translation of Hilbert's paper on sums of squares

Does anyone know if there is a French or English translation of Hilbert's paper on sums of squares: Ueber die Darstellung definiter Formen als Summe von Formenquadraten?

**2**

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118 views

### Computing harmonic sum [closed]

I want to show the following equalities for harmonic sum
$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$
Any idea?

**4**

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**1**answer

91 views

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

Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that ...

**5**

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220 views

### polynomial maps from reducible plane curves to conics

It is classically known that every smooth plane quartic curve $C$ can be represented by an equation $q_1 q_3 = q_2^2,$ with $q_j\in\mathbb{C}[X,Y,Z]$, $1\leq j\leq 3$ quadratic forms, and the same is ...

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177 views

### On sum of squares representations of primes

We know $p=1\bmod 4$ has an unique representation via $x^2+y^2=p$ where $x,y\in\Bbb N$ holds.
There are other unique form representations as well.
Suppose $p$ is a prime and we have coprime $\alpha,\...

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309 views

### what is this sum of squares of algebraic functions?

This question is inspired by the MO query here, although it has no direct implications.
Define the family of polynomial functions
$$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$
and ...

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686 views

### Efficient method to write number as a sum of four squares?

Wikipedia states that there randomized polynomial-time algorithms for writing $n$ as a sum of four squares
$n=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$
in expected running time $\mathrm {O} (\log^{2}...

**4**

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**1**answer

162 views

### Ensuring that a sum of squares is non-zero (in a finite field)

The following problem bears some similarity to the Additive Basis Conjecture [ALM91,JLPT92] saying (in characteristic $3$) that there is an absolute constant $N$ such that for any positive integer $m$,...

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450 views

### When is $ \sigma(n!-1) $ a perfect square?

I am looking for pairs of positive integers $(m,n)$ such that $ \sigma(n!-1) =m^2$, where $\sigma$ is the sum of divisors function. Examples occur with $(m,n)=(12,5),(1,2)$.
Question: Are there ...

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**1**answer

285 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 ...

**0**

votes

**1**answer

311 views

### Number of fixed points in Zagier's involution (Fermat's Theorem) [closed]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...

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126 views

### The Linnik problem for dimension $2$

For $N$ an integer, let
$$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$
For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...

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507 views

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

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...

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266 views

### Lower bound for the number of representations of integers as sum of squares

Let $k\geq 4$. As usual, let $r_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r_k(n) \gg n^{\frac{k}{2}-1-\...

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172 views

### Sums of hermitian squares in free abelian group algebras and real positive semidefinite matrices

A little context for the following question, first. As Noah Stein notes in a comment below, the present question is closely related to the free semialgebraic geometry studied by Helton and his ...

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**1**answer

499 views

### Realization of numbers as a sum of three squares via right-angled tetrahedra

De Gua's theorem
is a $3$-dimensional analog of the Pythagorean theorem:
The square of the area of the diagonal face of a right-angled tetrahedron
is the sum of the squares of the areas of the other ...

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122 views

### About the properities of sum of powers of items in a polynomial

Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials:
\begin{eqnarray*}
f_2&=&a_1^2x^2+\cdots+a_{p-...

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413 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 $...

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371 views

### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...

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203 views

### Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?

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721 views

### Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...

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387 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(\...

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1k views

### Many representations as a sum of three squares

Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...

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145 views

### Representation as sum of squares

Given $N\in\Bbb N$ such that $\prod_{i=1}^mp_i=N$ with $p_i$ being similar sized primes such that $p_i\neq p_j$ if $i\neq j$ where $m\in[1,\log\log N]$, consider $$r_4(N,[a,b])=|\{\alpha^2+\beta^2+\...

**6**

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**4**answers

316 views

### Application and usage of representation of integers as sum of powers?

We know that there are many articles and manuscripts from the ancient to date talking about representation of integers as sum of squares, cubes etc. I would like to know what is it the usage and ...

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**1**answer

159 views

### How many finite subsets in $\mathbb{Z}^d$ have a given sum of squares?

Let $|\cdot|$ denote the usual norm in $\mathbb{Z}^d$. Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum_{z \in S}|z|^2$. Given $m \in \mathbb{N}$, what is the size of $\varphi^{-...

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462 views

### Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv −1 \mod p$. Is there a possibility to say ...

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178 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 ...

**3**

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426 views

### Generating Function of distinct way of partitioned square sums of positive integers

Let's define a function $p_2(n)$ that it is total distinct way to write $n$ positive integer as sum of square of positive integers. For example, $9$ can be partitioned as sum of squares in 4 distinct ...

**3**

votes

**1**answer

486 views

### Sum of two squares and implication of Bunyakovsky conjecture

Bunyakovsky's conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...

**6**

votes

**1**answer

189 views

### For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?

For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...