The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
52 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 ...
13
votes
1answer
413 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 ...
2
votes
2answers
187 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-\...
2
votes
0answers
124 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 ...
13
votes
1answer
392 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 ...
6
votes
0answers
108 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-...
11
votes
2answers
374 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 $...
11
votes
0answers
341 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 ...
1
vote
2answers
178 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?
6
votes
1answer
261 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 ${\mathbb R}^2$). ...
7
votes
2answers
328 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(\...
12
votes
2answers
657 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 ...
0
votes
0answers
126 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
votes
4answers
278 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 ...
2
votes
1answer
149 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^{-...
1
vote
1answer
289 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 ...
1
vote
0answers
97 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
votes
0answers
290 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
1answer
440 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
1answer
159 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 ...
0
votes
1answer
191 views

Longest sequence of sum of distinct squares [closed]

I want to find longest sequence of distinct squares that $\alpha_{_1}$ + ... + $\alpha_{_n}$ is given number. In particular I want to find largest square in that sequence. I've tried use Lagrange's ...
6
votes
2answers
298 views

Divisor sums over values of binary forms of primes

Let $\tau$ be the divisor function, that is $$ \tau(n)=\sharp\{d \in \mathbb{N}, d|n\}. $$ I was wondering if anyone has ever proved an asymptotic estimate for the sum $$S(x):=\sum_{p,q\leq x}\tau(p^...
24
votes
1answer
520 views

Distribution of $a^2+\alpha b^2$

It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that ...
2
votes
0answers
192 views

Sum of Squares and Harmonic Functions

Let $a_0, a_1, \dots a_k$ be non-negative reals. Any homogeneous polynomial $p$ of degree $2k$ in $\mathbb{R}^{d}$ can be decomposed as $$ p(x)=\sum_{i=0}^{k}c_i|x|^{2(k-i)}p_{2i}(x) $$ for some $c_i$...
3
votes
1answer
132 views

Sum of Squares Length of a Product

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...
6
votes
2answers
658 views

Would such polynomial identity exist? (related to sum of four squares)

Let $f_1,f_2,f_3,f_4,f_5 \in \mathbb{Q}[x]$ be linear and coprime and not all constant. Is it possible $ f_1^2+f_2^2+f_3^2+f_4^2=f_5^2$? I suppose the answer is negative. If this is possible, ...
6
votes
2answers
573 views

Hurwitz integers represented as sums of two squares of Hurwitz integers

I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a ...
6
votes
1answer
559 views

On permuted sum of squares of primes in a list

We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\...
6
votes
1answer
446 views

Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$

The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it: Problem 19.5 (p. 224) ...
1
vote
2answers
230 views

Impossible Range for Minkowski-Like Sum of Squares

Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of $Mx^2-N(a^2+b^2+c^2)$, is there any such z which is "deceptive", meaning that it ...
2
votes
2answers
276 views

Four-Square Theorem for Negative Coefficient

What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be ...
4
votes
2answers
452 views

Numerical coincidence?

(Nobody's answered this one on stackexchange after several days.) My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right ...
7
votes
2answers
984 views

The four-square theorem from the Gauss-Legendre three-square theorem

I've been studying some proofs of the four-square theorem. Some of them are pretty clear. However, I came across a statement that the four-square theorem can be easily derived from Gauss-Legendre ...
11
votes
2answers
794 views

Sums of Squares

Every prime $p = 4k + 1$ can be uniquely expressed as sum of two squares, but for which integers $x$ is $x^2 + y^2 =$ some prime $p$? Stated differently, does the square of every positive integer ...
0
votes
0answers
206 views

Confirm/refute $f(x)$ where $f(x) = $x-th Mersenne prime ($M_p$) where $x$ is [1,2,3,4,5,6,7…]

While tinkering with numbers, I found $n=f(x)$ where $n = \sigma(\sigma(n)-n)$ and $x \neq p$ where $p$ is a Mersenne prime exponent, but I need help input in regard to improving accuracy. First off, ...
6
votes
3answers
1k views

Realizing proper pure octonions as conjugates

Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...
0
votes
0answers
103 views

maximum of the sum of polynomials

Hi I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show ...
1
vote
0answers
87 views

Information about mutant Leech lattice related to smallest perfect squared square

What happens if we follow the construction of the Leech lattice but replace the relation $\displaystyle \sum_{n=1}^{24} n^2 = 70^2$ with the smallest perfect squared square? Explicitly, if we set ...
6
votes
1answer
385 views

An S-lemma for polynomials of degree 4 in three variables

Might the following be true: Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the ...
3
votes
2answers
726 views

Integer partition and sum of squares

Hello, The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics) For all integers $n\geq 2$ denote by $\mathcal{P}...
4
votes
0answers
252 views

Product on representations of an integer by a quadratic form?

Define the quadratic form $$Q(z_1,z_2,z_3,z_4) = 13 + \sum_{i=1}^4 (10+i)z_i +5 \sum_{1 \le i \le j \le 4} z_iz_j.$$ Then, $r_Q(n) := \left|\{(z_1,z_2,z_3,z_4) \in \mathbb{Z}^4 : Q(z_1,z_2,z_3,z_4) ...
2
votes
1answer
301 views

Question on Sums of Squares

Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements ...
9
votes
1answer
700 views

Representations by positive definite binary quadratic forms

It's known that the number of representations of an integer $k$ by sum of two squares is $$ 4\;\sum_{d|k}\left(\frac{-4}{d}\right) $$ or $$ 4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= 4(d_1(k)...
0
votes
2answers
707 views

Algorithm for calculating the sum-of-squares distance of a rolling window from a given line function

Given a line function $y = ax + b$, it is easy to calculate the sum-of-squares distance between the line and a window of samples $(1, y_1), (2, y_2), ..., (n, y_n)$ (where $y_1$ is the oldest sample ...
13
votes
1answer
1k views

Legendre and sums of three squares

The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed to have given a proof ...
5
votes
1answer
444 views

A 'generalized Four Squares Theorem'?

The $4$-dimensional lattice $\mathbb{Z}^{4}$ has vectors of length $\sqrt{n}$ for any positive integer $n$ by the Four Squares Theorem, but this need not be true for higher-dimensional integral, ...
4
votes
1answer
625 views

quartic diagonal as a sum of squares of quadratic forms

I would appreciate if someone can point out to the literature related to characterizing the set of all different ways to write real quartic diagonal $\sum \limits_{k=1}^n x_k^4, x \in \mathbb{R^n}$ as ...
2
votes
1answer
726 views

On a claim of Ramanujan in his “Lost Notebook”.

As I was flipping through the scanned version of Ramanujan's "Lost Notebook" in our library, I came across a result which caught my attention. And as any excited teenager would do, I immediately ...