Questions tagged [sums-of-squares]

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

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53 views

Largest $k$ such that $(p-k) = \lceil \sqrt{(p-2k) p} \rceil$

Assume $p \in \mathbb P.$ Assume $0<p-2k<p$ and the next square larger than $p(p-2k)$ is $(p-k)^2$. It is trivial to show that $p(p-2k)+k^2$ is a square. Simply $p(p-2k)+k^2 = (p-k)^2.$ ...
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1answer
51 views

Maximum length of a representation of a number as an alternating sum of squares

Define a function $$\mathscr R: \mathbb N \to \mathbb N, \ \ \mathscr R(n) = \lceil \sqrt{n} \rceil ^2 -n.$$ IE. the distance of $n$ to the next prefect square. Sequence A068527 on OEIS. If $\mathscr ...
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4answers
1k views

Represent an integer as a sum of n non-consecutive squares

Is it possible to decompose an integer into a sum of n unique squares ,even though they are not necessarily consecutive. For instance, How would I obtain the sequence 1*1 + 7*7 + 14*14 + 37*37 given ...
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1answer
47 views

Find $2^{2}+4^{2}….(2n)^{2}$?

I tried subtracting $1^{2}+3^{2}….n^{2}$ from $2^{2}+4^{2}….2n^{2}$, but that didn't work. I know the answer is
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2answers
2k views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
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2answers
55 views

Non negativity condition for quartic polynomials?

Say I have a quartic polynomial $f(x) = ax^4 + bx^3 + cx^2 + d$. I am told that $f(x)$ is nonnegative iff it can be expressed as a sum of squares as follows. $f(x) = \sum_{i=1}^4 q_i(x)^2$. As an ...
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5answers
173 views

Relationships between the elements $(a,b,c,d)$ of a solution to $A^2+B^2+4=C^2+D^2$

I have reduced a certain equation (in positive integers) to the equation $$A^2 + B^2 + 4 = C^2 + D^2. \quad(\star)$$ Assume the positive integers $(a,b,c,d)$ are any solution to $(\star)$. Are there ...
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2answers
335 views

Problem Solving Question? Sum of the squares

The sum of the squares of two numbers is 247 and the product of the two numbers is 21. How would I find all possible values for the sum of the two numbers?
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3answers
173 views

Does this equation have positive integer solutions?

The only solution I can find for $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ is $a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$. Are there any positive integer solutions? Any where none of $a,b,c,d,e,f$ are ...
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1answer
132 views

Azuma's inequality: Expected sum of differences

I am looking for an extension of Azuma's inequality which involves the expected sum of squared differences. In particular, recall that Azuma's inequality states \begin{align*} \Pr[X_n-X_0 \geq a] \leq ...
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2answers
292 views

squares which are not the sum of a square and twice a triangular number

I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ ...
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4answers
3k views

sum of 4 squares

Is there any natural number $A$ which cannot be written as: $$A=W^2+X^2+Y^2+Z^2$$ where $W,X,Y,Z \in \mathbb N \cup 0$ I was considering the fact that $a^2+b^2 \not = 1 \mod 4$ and was attempting to ...
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2answers
247 views

Prove that no $n,m, 0<n<m$ exist such that $m^2 +mn+n^2$ is a square number

Prove or disprove the claim that there are integers $n,m, 0<n<m$ such that $m^2 +mn+n^2$ is a perfect square.
18
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1answer
534 views

All elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as sum of a square and a cube?

Is it true that all elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as the sum of a square and a cube? Example: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 \...
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0answers
82 views

A four-digit square is of the form $aabb$. What's $a^2+b^2$? [duplicate]

The 5772 Ulpaniada included the following question: Consider a four digit square number (a number which is the square of a whole number).Its digit notation is $aabb$ (the thousands digit is $a$, ...
7
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1answer
522 views

Algorithm to find solution to $ax^2 + by^2 = 1$ in a finite field

Let $\mathbb{F}$ be a finite field, and let $a,b \in \mathbb{F}$ be given, subject to $a\ne 0, b \ne 0$. Consider the equation $$ax^2 + by^2 = 1.$$ It is guaranteed that there exists a solution to ...
7
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1answer
340 views

Weighted sum of squares, in a finite field

Let $\mathbb{F}$ be a finite field. Let $a_1,\dots,a_n \in \mathbb{F}$ be given. I want to know whether there exists $x_1,\dots,x_n \in \mathbb{F}$ such that $$a_1 x_1^2 + a_2 x_2^2 + \dots + a_n ...
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0answers
134 views

Showing a Hermitian preserving map is not necessarily positivity preserving?

Say I have a linear map, which is not positivity preserving $$\phi: \mathscr H \to \mathscr H$$where $\mathscr H$ is the set of $n \times n$ Hermitian matrices. Then does there exist a positive ...
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1answer
41 views

Establishing an Upper Bound for a Curious Function

Suppose I have a sequence of positive real numbers $a_1, \, a_2, \, \dots \,a_n$ such that the following is satisfied: $\sum \limits_{i=1}^{n}a_i = 1$ I am trying to find the smallest value of $L$ ...
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1answer
1k views

Dot product of the column vectors from a matrix and their transposes through matrix multiplication

I have a matrix with data, every dataset is a column vector in my matrix. I want to know the dot product of the transpose of each column vector with the original column vector. If I transpose the ...
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1answer
39 views

Arrangement of the following term

How $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2m}}$$ can be rewritten as $$\sum_{k=1}^{\infty}\frac{(-1)^{2k-2}}{(2k-1)^{2m}}+\sum_{k=1}^{\infty}\frac{(-1)^{2k-1}}{(2k)^{2m}}$$ ???
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2answers
773 views

Representing an Integer as a Sum of at Most $k$ Triangular Numbers

What is the smallest $k$ such that every $n \in \mathbb{N}$ can be represented by a sum of exactly $k$ triangular numbers? For the sake of simplicity, I will assume $0$ is a triangular number. I've ...
7
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3answers
295 views

$ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $

I saw this problem recently and found an elegant solution to it, and was curious to see if anybody would think of something else. Nice solutions to nice problems are fun to see! Problem: Prove ...
4
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3answers
173 views

Solving a function for square numbers

Essentially I'm curious; could a perfect square($x$ squared) be less than the sum of all lesser perfect squares by a perfect square, and if so, what would the smallest solution be. Take $36$ for ...
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1answer
198 views

Why does sum of squares appear in so many mathematical applications?

I have some little background in statistics, where in many applications summing of squares is an important calculation. Recently, I came across a mention that summing squares is involved in ...
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1answer
4k views

A prime congruent to 3 modulo 4 & sums of squares

Prove: If $p$ is a prime where $p \equiv 3 \pmod{4}$ then $p$ can't be written as the sum of two numbers squared. I attempted by contradiction, supposing that $p=a^2 + b^2$ where $a,b$ are integers....
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0answers
1k views

How to prove Sum of squares of $n$ numbers is unique

I was solving a programming problem where I needed to prove that sum of squares of $n$ numbers is unique. i. e. if calculate the sum of squares of equal number of positive integers then if they are ...
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0answers
477 views

Optimization: Minimizing Quadratic Vector Valued Functions

I have a vector valued function $F: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$, which consists of quadratic taylor approximations. So one could say that function F consists of stacked approximations ...
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2answers
106 views

Solving for $a,b,c,d$ where $a^2 + b^2 + c^2 + d^2 = 630^2$

How could one solve for $a,b,c,d$ where: $$a^2 + b^2 + c^2 + d^2 = 630^2,\ a>b>c>d$$ $a,b,c,d$ squared is equal to the square of $630$, and $a$ is larger than $b$, and so forth. $a,b,c,d$ ...
4
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2answers
181 views

For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 \leq i \leq n$?

I need to prove this result for something I am working on: For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 \leq ...