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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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 ...
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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 ...
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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 $a,b\in\mathbb{Z}$?
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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 ...
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 ...
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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 $a < b$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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^...
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 =...
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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?
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.
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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?
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 ...
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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 ...
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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 ...
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$\sum_{n=1}^{2017}\left(\left((n+2)^4\bmod{(n+1)^4}\right)\bmod{4}\right)$

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^...
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Explain the following derivation

Can you explain the derivation in the given image? Which steps lead to the incorrect conclusion?
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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.
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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 ...
$\forall n \in \mathbb{N}_{>5}\implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$
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 ...