# 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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### 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^...
152 views

### Solutions to a system of three equations with Pythagorean triples

Is there any solution to this system of equations where $x,y,z,s,w,t\in\mathbb{Z}$, none are $0$. \begin{align*} x^2+y^2=z^2\\\ s^2+z^2=w^2\\\ x^2+t^2=w^2 \end{align*} EDIT: Thank you zwim for the ...
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### Prove that $x^2+2y^2+3z^2=10a^2$ has no integer solutions aside from all of them being 0

I got this equation while I was trying to solve a certain math Olympiad problem. I tried modulus and whatnot, but I haven't got anywhere. Is there a way to prove this?
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### Finding Pythagorean triplet given the hypotenuse

I have a number $c$ which is an integer and can be even or odd. It is the hypotenuse of a right angled triangle. How can I find integers $a,b$ such that $$a^2 + b^2 = c^2$$ What would be the ...
177 views

### In ℕ⁺, can the sum of three squares equal the sum of two squares?

Are there any examples where: $a² + b² + c² = p² + q²\qquad {a, b, c, p, q ∈ ℕ⁺}\tag{1}$ If not, can $(1)$ be disproven?
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. ...
455 views

9k views

### Diophantine equation $a^2+b^2=c^2+d^2$

I was reasonably certain I've seen this before, but I was wondering how to solve the Diophantine equation $$a^2+b^2=c^2+d^2$$ I tried a web search and found nothing on this one. I'm trying to avoid ...
186 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 ...
93 views

### The sum of an infinite series containing a finite series in each denominator [duplicate]

Evaluate $$\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{6^2}+\frac{1}{10^2}+\cdots+\frac{1}{\left[\frac{k(k+1)}{2}\right]^2}+\cdots$$ to $\infty$, where $k$ is the $k$th term of the series. Using ...
1k views

### How can I write $1105$ as the sum of two squares other than $33^2 + 4^2$? [duplicate]

How can I write $1105$ as the sum of 2 squares other than $1105 = 33^2 + 4^2$? Could someone explain to me a procedure for doing this? I know that it has at least 2 other representations as a sum of ...
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### Arithmetic Derivative on sum of two perfect squares

Let $n,m \in \mathbb N$ and $n$ even, $m$ odd. If we take there squares and add them $n^2+m^2$, are there examples when we take the arithmetic derivative of the sum: $(n^2+m^2)' \equiv 0 \mod 4$ ?
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### How to work out sum of Square Number from given numbers?

Which of the following cannot be written as the sum of two distinct square numbers? A.106 B. 109 C. 112 D. 117 What would be the correct answer here and can someone explain in detail please.
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### Pattern in Squared Numbers and their Digit Sum

So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
51 views

### Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$. Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$. Is that ...
9k views

### Natural number which can be expressed as sum of two perfect squares in two different ways?

Ramanujan's number is $1729$ which is the least natural number which can be expressed as the sum of two perfect cubes in two different ways. But can we find a number which can be expressed as the sum ...
49 views