# Questions tagged [sums-of-squares]

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### Sum of four squares from sum of $k\geq5$ squares - exact version $\mathsf{II}$

Every natural number is sum of $4$ squares and every natural number bigger than $169$ is sum of five squares. Suppose $n>169$ and $x'x=n$ at a vector $x\in\mathbb Z^5$ then can we find two ...
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### 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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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 $A \in M_n(... 2answers 377 views ### Extension of Dickson's theorem on integers of the form$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 \... 3answers 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 ... 1answer 314 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 ... 2answers 281 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 ... 1answer 557 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 ... 0answers 515 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: ... 0answers 47 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). ... 1answer 302 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,$$where a_1,a_2,a_3 are ... 1answer 648 views ### Efficient sum of squares decomposition Sum of 4 squares decomposition is the well-known result. I'm interested only in negative/non-negative separation with focus on efficiency and large numbers. I'm looking for alternatives or extensions ... 0answers 512 views ### Does every integer n>1 have the form a^2+b^2+3^c+5^d with a,b,c,d nonnegative integers? Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: ... 0answers 424 views ### If ab^2 is a sum of three squares, then so is a. How to see it quickly? Here a, b are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way? 1answer 69 views ### For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials Let \mathbb R_d[t] be the set of univariate polynomials in the variable t of degree d, and S be the set of elements of \mathbb R_d[t] that are nonnegative on [0, 1]. Does the following ... 1answer 356 views ### Every odd integer greater than 1 is of the form a+b with a^2+b^2 being prime Let m be an odd integer greater than 1. Is it true that there are positive a, b such that m=a+b and a^2+b^2 is a prime number? It seems that for every odd m there are many (a,b)\in \... 0answers 165 views ### Translation of Hilbert's paper on sums of squares Does anyone know if there is a French or English translation of Hilbert's paper on sums of squares: Ueber die Darstellung definiter Formen als Summe von Formenquadraten? 0answers 126 views ### Computing harmonic sum [closed] I want to show the following equalities for harmonic sum$$\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$$Any idea? 1answer 100 views ### Specific quaternary quartic that is positive semi-definite but not sum of squares Does there exist a quaternary quartic f (a form in \mathbb{R}[x_1,x_2,x_3,x_4] of degree 4), which is positive semi-definite (f \geq 0 on \mathbb{R}^4) but not a sum of squares, such that ... 1answer 231 views ### polynomial maps from reducible plane curves to conics It is classically known that every smooth plane quartic curve C can be represented by an equation q_1 q_3 = q_2^2, with q_j\in\mathbb{C}[X,Y,Z], 1\leq j\leq 3 quadratic forms, and the same is ... 2answers 322 views ### what is this sum of squares of algebraic functions? This question is inspired by the MO query here, although it has no direct implications. Define the family of polynomial functions$$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$and ... 3answers 1k 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}...
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 $m$,...
### 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 ...