# 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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### Can the sum of the first n squares of primes be a perfect square?

So far, all I have found is for $n=1$, for which the square is 4. (Rather trivial) To give some context, I recently watched the Numberphile video Squared Squares. Basically, a squared square is a ...
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### How can I prove the sum of squares and the sum of cubes with binomial coefficients? [closed]

My main problem is starting. I can't "see" anything that might give me an idea to find a relationship between these two things Thank you :)
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### Prove or disprove a claim involving Pythagorean primes

As we know, every Pythagorean prime $p$ is expressible as $p=x^2+y^2$ such that $x$ is an odd integer and $y$ is an even integer. Prove or disprove: $x$ is a quadratic residue modulo $p$.
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### Remainder of $\sum_{x=1}^{312} x \times x!$ divided by 2016 [closed]

I have this question: What is the remainder of $$\sum_{x=1}^{312} x \times x!$$ (or just simply) $$(1! \times 1) + (2! \times 2) + (3! \times 3) + \dots + (312! \times312)$$ Divided by ...
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### Two times the sum of two squares is a sum of two squares

For all $a,b\in\mathbb Z$ there are $c,d\in\mathbb Z$ such that $2(a^2+b^2)=c^2+d^2$. The conjecture is tested for $a^2+b^2<1,000,000$ but I have problems with proving it.
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### Find the value of: $\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$ [closed]

Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$ I need help solving this question. Every reply is appreciated. Thanks!
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### Find the value of $\frac {1} {1^2 + 1} +\frac {1} {2^2 + 2} +\frac {1} {3^2 + 3} +\frac {1} {4^2 + 4} ++ \dots + \frac {1} {2008^2 + 2008}$

I have this question: Find the value of: $$\frac {1} {1^2 + 1} +\frac {1} {2^2 + 2} +\frac {1} {3^2 + 3} +\frac {1} {4^2 + 4} ++ \dots + \frac {1} {2008^2 + 2008}$$ My attempt: I ...
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### A method to study the factorization of very big numbers

Is there a method to construct very big numbers in a given interval, with rectangular distribution, by selecting prime factors randomly? I want to study factorizations of the kind $a=b\cdot c$, where ...
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### Is $7m^2-3n^2$ a perfect square?

Is $7m^2-3n^2$ a perfect square for all positive integers $m,n$? I tried using double induction, but failed. Any other approach? By the way, is this related to fermat's theorem on representation of ...
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### Sum of two squares of an integer N, the simplest algorithm?

The algorithm is based on the following observation. An integer $b>a$ can be written as $b=a + k$ with a and k integers. So if we want to find a sum of 2 squares of an integer N, we start by ...
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### Find all prime numbers $y$ and $z$ if $x$ is an integer and $x^2-y^2-z^2=2017.$

Find prime numbers $y$ and $z$ if $x$ is an integer and $x^2-y^2-z^2=2017.$ $x^2-y^2-z^2$ is not factorable, so what should I do? My mind is blank here. Any help is appreciated.
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### Algorithm to Divide the number into 4 square part

I am having a number N i want to divide this number into sum of square contains upto 4 numbers. ...
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### Summation formula proof

For $k$ and $l$ $\in n$ how can I prove the following : $$(\sum_{k=1}^{k=n}\pi_{k}Y_{k}-\sum_{k=1}^{k=n}Y_{k})^2=\sum_{k=1}^{k=n}\sum_{l=1}^{l=n}Y_{k}Y_{l}(\pi_{k}-1)(\pi_{l}-1)$$ . I have really ...
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### Polynomial equal to sum of squares of polynomials [duplicate]

Given a nonnegative polynomial $p(x)$ on $\mathbb{R}$, does there exist some $k$ such that for some polynomials $f_1,\ldots ,f_k$ we have $p(x)=\sum_{i=1}^k(f_i)^2$? I think yes, because of the ...