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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5
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2answers
522 views

How many sub-square matrices does a square matrix have and is there a simple formula for it?

Consider an $n \times n$ matrix $M$. I want to find the determinant for ALL sub-square matrices of $M$. There may be a better way but my method is to find all sub-square matrices and check them ...
0
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2answers
48 views

Sum of the Squares of the First n square Numbers is not a perfect square number

In other words, $x(x+1)(2x+1)=6y^2$ has no nontrivial integral solutions. I thought this is a well-known result, but surprisingly could not find a recorded (easy) proof. Can someone provide a proof ...
1
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2answers
55 views

consecutive integers that are not the sum of 2 squares. [closed]

Are there any $n\in\mathbb{N}$ such that no element $k\in\{n,n+1,n+2,...,n+2017\}$ can be expressed as $a^2+b^2$ for some $a,b\in\mathbb{Z}$?
1
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0answers
166 views

How to interpret this visual proof for Archimedes' derivation of Sum of Squares?

I'm interested in knowing how was the first closed form solution of the sum of squares derived (for historical context/curiosity). I stumbled across an MAA Page that provides a visual proof, without ...
5
votes
3answers
122 views

If $x^2-dy^2 = -1$ has a solution in $\mathbb{Z^2}$, then $d$ is the sum of two coprime squares.

I want to prove: Let $d\in\mathbb{N}$ be square-free. If $x^2-dy^2 = -1$ has a solution in $\mathbb{Z^2}$, then $d$ is the sum of two coprime squares. I've already shown, that the equation has no ...
0
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1answer
75 views

How many variations of new primitive Pythagorean triples are there when the hypotenuse is multiplied by a prime?

When I was looking at all the Pythagorean triples from 1 to 1000, I noticed a certain trend which I cannot seem to explain. Given $a, b, c$ are integers where $c$ is the hypotenuse, and $a < b$, ...
2
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2answers
67 views

How to find the sum of the Series $\sum_{n \in \Bbb{Z}} \vert c_n \vert^2$ which terms are the special integral representations

For $n \in \Bbb{Z}$, define $$ c_n=\frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} e^{i(n-i)x} dx$$ where $i^2=-1$. Then $\displaystyle\sum_{n \in \Bbb{Z}} \vert c_n \vert^2$ equals........? a) $\cosh(\...
1
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1answer
41 views

Confirmation of Proof: $\forall n \in \mathbb{N}^*, \ p = 4261 + 2308n : \text{prime} \Rightarrow p = \{a^2 + b^2 \mid \gcd(a, b) = 1\}$

I developed a conjecture which I would like to confirm whether or not it holds truth for all members $n$ of the set of $\mathbb{N}^* = \mathbb{N} \ \cup \ \{0\}$. $$\forall n \in \mathbb{N}^*, ...
3
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1answer
21 views

Is there a pattern defining the existence of root integer distances in an isometric grid?

In a standard square grid pattern the distances to integer root locations is simply the sum of two squares. We find that these distances have $\sqrt1, \sqrt2, \sqrt4, \sqrt5, \sqrt8, \sqrt9, \sqrt10, \...
4
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1answer
148 views

Can this be continued forever?

We have: $$1=1^2$$ and $$25=5^2=3^2+4^2$$ and $$441=21^2=20^2+4^2+5^2$$ So, for $k=1,2,3$ we have a $k$-digit number that is a perfect square and a sum of $k$ different non-zero perfect squares. ...
2
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2answers
88 views

Is there an infinite number of numbers like $1600$?

My reputation is at this moment at $1600$. I did some experimenting with $1600$ and obtained the following: Evidently, it is a perfect square $1600=40^2$ Also, it is a hypothenuse of a Pythagorean ...
0
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1answer
39 views

Calculating the Sum of Squares between in ANOVA table

I can't figure out where I have gone wrong here. There are 3 independent samples all of which have 12 observations, which give values: $\Sigma x_a = 4913, \Sigma x_b = \Sigma x_c =5177, \Sigma x_a^2 ...
-1
votes
2answers
192 views

sum of square derivative

What is the partial derivative of the following expression with respect to $U_i,V_j$ and M, respectively: $$L=\sum_{i}^m \sum_{j}^n(P_{ij} - g(U_i^T M V_j))^2 $$ where $$ U \in R^{d*m} , V \in R^{d*...
4
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0answers
151 views

Iterations of $x^2 + y^2$

We construct a sequence $S$ of distinct positive integers as follows 1) the sequence $S$ starts as $1,2,3$ 2) If $x,y$ are in the sequence , then $x^2 + y^2 $ is also in the sequence. 3) the ...
2
votes
1answer
116 views

More of this type $T_n^2+T_{n+1}^2+T_{n+2}^2=X^2?$

Given triangular numbers, $$T_n:= {n(n+1)\over 2} = 1,3,6,10,15,...$$ Let $T_n, T_{n+1}$ and $T_{n+2}$ be three consecutive triangular numbers. My question is: Are there more of $T_n^2+T_{n+1}^2+...
0
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4answers
82 views

Integer solutions to $a^{2}+b^{2}+c^{2} = d^{2}+e^{2}+f^{2}$, where $a, b, c, d, e, f \neq 0$

How does one approach something like this? Is there an equivalent Legendre's three-square theorem for the sum of three squares in two different ways? It seems like the only way to approach it would ...
0
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0answers
34 views

Is there reference sheets or techniques to writing numbers as sums of squares?

For example if I wanted to write $x$ as the sum of $n$ squares in $m$ different ways, is there something I can just look up? Also I have other constraints with what I am working with, if I knew $x$ I ...
1
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1answer
38 views

Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$)

Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$). I learned about quadratic residue and sums of ...
0
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1answer
491 views

sum of 3 squares [duplicate]

I want to show that any integer of the form $n=4^m(8k+7)$ with $m,k\ge0$ cannot be expressed as a sum of 3 squares. The case for $m=0$ is easy to prove since the sum of 3 squares cannot be $\equiv7\...
3
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1answer
62 views

Show that $ \# \{ (a,b,c,d) \in \mathbb{Z}^4 : a^2 + b^2 + c^2 + d^2 = m\} \asymp m $

I found this result mentioned in passing in a number theory paper. It looks almost self-evident: $$ \# \{ (a,b,c,d) \in \mathbb{Z}^4 : a^2 + b^2 + c^2 + d^2 = m\} \asymp m $$ It is stated without ...
2
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1answer
24 views

if $\sum_{k=1}^nx_{k}=\sum_{k=1}^{n}x_{k}^2=n$ then forall k, $x_k=1$

Let $(x_{k})_{1\leq k\leq n}$ be a set of real numbers such as $\sum_{k=1}^nx_{k}=\sum_{k=1}^{n}x_{k}^2=n$ I need to give proof that $\forall k\in${$1, 2, ..., n$} $x_k=1$ I spent hours trying to ...
0
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2answers
861 views

Representing a given number as the sum of two squares.

There are four essentially different representations of $1885$ as the sum of squares of two positive integers. Find all of them. I am guessing the must be some nice way of solving $z= x^2 + y^2 $ ...
2
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1answer
50 views

Looking for a reference about a problem on the number of representations of an integer as a sum of two squares

Few years ago I came accross a paper, or maybe a solution of a problem from a journal (possibly AMM or something like that) in which the following result was proved: The smallest positive integer ...
4
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2answers
80 views

Is there a geometric realization in integer-sided squares of $70^2 =\sum_{j=1}^{24} j^2 $?

I saw this in the NAdigest mailing list, and it was obviously suggested by $70^2 =\sum_{j=1}^{24} j^2 $: From: Gerhard Opfer gerhard.opfer@uni-hamburg.de Date: November 06, 2017 Subject: ...
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1answer
21 views

Nonnegative Polynomials Imply Nonnegative Probability Measure Product?

Let $f$ and $g$ be polynomials over $\mathbb R^n$. Let $X \subset \mathbb R^n$ be a compact, convex set (a polyhedron, specifically, if that makes a difference). Is the following true? Claim: If $f(...
0
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1answer
49 views

Five Trig Functions who's squares add to a constant?

I was trying to think of a set of five trig functions in which the first trig function is multiplied by some constant a, the second by a different constant b, the third by a different constant c, the ...
0
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1answer
253 views

Riemann Sums as definite integrals

I looked at all the resources for Riemann Sums for BC calculus and I could not find any that solved them like my teacher does. The question asks: Express the following Riemann Sums as definite ...
1
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1answer
39 views

Quadratic fit using sum of least squares without matrices

I've been hunting around for examples using the sum of least squares + partial derivative method to fit a polynomial to a set of points but am completely stuck. All the examples I've found involve ...
4
votes
2answers
136 views

Minimum over Probability Measures

Let $f$ and $g$ be polynomials in $\mathbf x \in \mathbb R^n$. Let $X$ be a compact subset of $\mathbb R^n$. Finally, Let $\mathcal M(X)$ be the set of probability measures over $X$. Can the ...
1
vote
1answer
1k views

How to compute SSR with just residuals and Xi?

How do we calculate SSR? I know SSE is the square of residuals all added together, but SSR is a subtraction between prediction for each observation and the population mean. Not sure how calculate SSR. ...
1
vote
1answer
78 views

Solving a Quadratic Ternary Form with Large Coefficients

I encountered this expression $$1215696x^2+566544y^2-103776z^2=0$$ which I've understood is called a "quadratic ternary form" and have been trying to find solutions $x,y,z\in\mathbb{Z}$ One can ...
3
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2answers
135 views

Find two arithmetic progressions of three square numbers

I want to know if it is possible to find two arithmetic progressions of three square numbers, with the same common difference: \begin{align} \ & a^2 +r = b^2 \\ & b^2 +r = c^2 \\ & a^...
2
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0answers
56 views

Arithmetic progression of four square number with the same common difference [duplicate]

I want to know, if is possible found the arithmetic progression of four square number, with the same common difference. \begin{align} \ & a^2 +r = b^2 \\ & b^2 +r = c^2 \\ & c^2 +r =...
0
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2answers
96 views

On sum of two squares

Can a prime of form $3\bmod 4$ ever divide an integer represented by form $a^2+b^2$ with $a,b$ being coprime and $b$ being even?
3
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3answers
128 views

How to prove that $441 \mid a^2 + b^2$

How to prove that $441 \mid a^2 + b^2$ if it is known that $21 \mid a^2 + b^2$. I've tried to present $441$ as $21 \cdot 21$, but it is not sufficient.
16
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4answers
1k views

Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$

Let $\alpha$, $\beta$, $\gamma$, $\delta$ be the roots of $$z^4-2z^3+z^2+z-7=0$$ then find value of $$(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$$ Are Vieta's formulas appropriate?
5
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3answers
273 views

Solve $32x^2 -y^2 = 448$

I am trying to find all integer solutions to the following equation: $$32x^2 - y^2 = 448$$ This is what I have tried so far: The equation describes a hyperbola, and so I try the usual trick of ...
12
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3answers
1k views

Can we prove a stronger claim?

Here it is asked whether for every $k>1$, there is a prime $p$ and $w_1,w_2,\cdots ,w_k>1$, such that $p$ divides $w_1^2+w_2^2+\cdots +w_k^2$, but none of the summands. Can we prove there is ...
5
votes
2answers
143 views

Primes dividing sums of squares but not dividing any of summands

I am inexperienced when it comes to number theory (even elementary one) so do not have an idea at this moment on how to solve this one. Let us go through some examples. For $k=2$ we can have prime ...
3
votes
0answers
67 views

Proof algorithm to find representations as sum of two squares

I saw in a book the following algorithm to find, given a prime $p\equiv 1 \pmod 4$, integers $a $ and $b $ such that $p=a^2+b^2$. Step 1 : Find an integer $0 <m <p$ such that $m^2\equiv -1\...
11
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1answer
190 views

Standalone proof of a conditional part of Lagrange’s Four-Square Theorem?

Lagrange’s Four-Square Theorem — a special case of the Fermat-Cauchy Polygonal Number Theorem (FCPNT) — states that every natural number can be written as the sum of the squares of at most four ...
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0answers
88 views

The sum of the squares of $5$ consecutive primes is again prime. True for infinite many quintuples?

Let $(p,q,r,s,t)$ be a quintuple of consecutive primes. Are there infinite many such tuples such that $$p^2+q^2+r^2+s^2+t^2$$ is prime again ? Motivation : $p^2$ is never prime $p^2+q^2$ is ...
0
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1answer
52 views

A simple explanation to this asymmetry?

Let $S_k^\delta=\{a+b\le k\mid(a,b)\in\mathbb N_+^2\wedge a^2+b^2\in\mathbb P\wedge a^2+b^2+\delta\in\mathbb P\}$, where $\mathbb P$ is the set of primes. If $\delta=2$ then the condition on $a^2+b^...
1
vote
5answers
65 views

$\sum_{n=1}^{2017}\left(\left((n+2)^4\bmod{(n+1)^4}\right)\bmod{4}\right)$

What's $$\sum_{n=1}^{2017}\left(\left((n+2)^4\bmod{(n+1)^4}\right)\bmod{4}\right)$$ What have I tried? $$(n+2)^4=n^4+8n^3+24n^2+32n+16$$ $$(n+1)^4=n^4+4n^3+6n^2+4n+1$$ Remainder: $$4n^3+18n^...
0
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1answer
40 views

Explain the following derivation

Can you explain the derivation in the given image? Which steps lead to the incorrect conclusion?
0
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1answer
43 views

Bivariate Form Sum of Squares

Let $F \in \mathbb{R}[x,y]$ be nonnegative and homogeneous of degree $2n$. Then it can be written as a sum of two squares.
0
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1answer
879 views

Put quadratic form into sum of squares

Is there a method or process that doesn't require a matrix to put quadratic forms into a sum of squares ? Two examples that I find extremely challenging. i) $q(x, y, z) = (x − y) ^2 + (y − z) ^2 − ...
7
votes
2answers
100 views

$\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P\}=\{k>1|\gcd(k,m+n)=1\}$

Conjecture: $\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P^{>2}\}=\{k>2|\gcd(k,m+n)=1\}$ if $m,n\in \mathbb N^+$ and $\gcd(m,n)=1$. This is a generalization of Any odd number is of form $...
4
votes
0answers
132 views

Even numbers has the form $a+b$ where $\frac{a^2+b^2}{2}$ is prime

Conjecture: All even numbers $n>2$ can be written $n=a+b$ where $a,b\in\mathbb N^+$ and $\frac{a^2+b^2}{2}\in\mathbb P$. Is tested for $n<10,000,000$. This conjecture is related to and maybe ...
4
votes
3answers
83 views

$\forall n \in \mathbb{N}_{>5}\implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$

Conjecture: $\forall n \in \mathbb{N}_{>5} \implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$ Tested $\forall n\leq 100,000$. Small exceptions: {1,2,3,5}. I would like to ...