# 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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### 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 ...
5answers
2k views

### Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
3answers
14k views

### Number of integer solutions of $x^2 + y^2 = k$

I'm looking for some help disproving an answer provided on a StackOverflow question I posted about computing the number of double square combinations for a given integer. The original question is ...
1answer
980 views

6answers
8k views

### Find all solutions: $x^2 + 2y^2 = z^2$

I'm use to finding the solutions of linear Diophantine equations, but what are you suppose to do when you have quadratic terms? For example consider the following problem: Find all solutions in ...
8answers
1k views

### Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.

For elements $a$ and $b$ in the ring $\Bbb{Z}$ prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$. I tried proving it but I just don't manage to. Maybe I am missing some basic claims in the ...
1answer
998 views

0answers
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### Can the sum of two squares be used to determine if a number is square free?

Mathword (http://mathworld.wolfram.com/Squarefree.html) stated that "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer."...
2answers
838 views

### Fermat's 2 Square-Like Results from Minkowski Lattice Proofs

Minkowski's Convex Body Theorem for lattices in the plane: Suppose $\mathfrak{L}$ is a lattice in $\mathbf{R}^2$ defined as $\mathfrak{L}=\{m\vec{v_1}+n\vec{v_2}:m,n\in\mathbf{Z}\}$, where $\vec{v_1}$ ...
0answers
296 views

### Can the sum of two squares be used to factor large numbers?

The method described below is different from Euler sum of two squares factorization method. The method below does not require 2 sum of two squares representations to factor a number. This post was ...
1answer
276 views

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 ...
3answers
784 views

### Show that $x^2+y^2+z^2=999$ has no integer solutions

The question is asking us to prove that $x^2+y^2+z^2=999$ has no integer solutions. Attempt at a solution: So I've noticed that since 999 is odd, either one of the variables or all three of the ...
3answers
248 views

### Let $p$ be a prime so $p\equiv3\pmod4$. If $p|a^2+b^2$, then $p|a,b$

Let $p$ be a prime so $p\equiv3\pmod4$. If $p| a^2+b^2$, then $p| a,b$ How do I prove this small theorem? I know that it's quite useful. Are there other small theorems like this one? I am mostly ...
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 ...
2answers
243 views

### How to derive the formula for the sum of the first $n$ perfect squares? [duplicate]

How do you derive the formula for the sum of the series when $S_n = \sum_{j=1}^n j^2$? The relationship $(n, S_n)$ can be determined by a polynomial in $n$. You are supposed to use finite differences ...
2answers
146 views

### Which numbers less than 5 billion have the most representations as the sums of two squares?

1842675848 has four representations as the sum of two squares: 1842675848 = 3518^2 + 42782^2 = 5458^2 + 42578^2 = 11338^2 + 41402^2 = 19702^2 + 38138^2. It got me to wonder which numbers in this ...
1answer
812 views

### Numbers as sum of distinct squares

Yesterday Polish Olympiad of Information Science ended, one of the questions was purely mathematical, Squares (PL). In the task, we have defined square ...
4answers
131 views

### Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$

After a lot of messing around today I curiously observed that $a^2+b^2$ is only divisible by 3 when both $a$ and $b$ contain factors of 3. I am trying to prove it without using modular arithmetic (...
3answers
2k views

### Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even.

Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even. I'm stuck on the best way to get this started. My thinking is that I could use cases. i.e. Case 1: a is even and b is ...
1answer
80 views

### If $p\equiv 3\pmod{4}$ and $p\mid x^2+y^2$, prove $p\mid x,y$.

I have to prove that if $p$ is a prime number of the form $p = 4n - 1$, $n\in N$ and $x^2+y^2\equiv 0\pmod{p}$, then $x\equiv 0\pmod{p}$ and $y\equiv 0\pmod{p}$. I have gone about this as follows and ...
2answers
803 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 $z= x^2 + y^2$ ...
1answer
30 views

### $a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$, deduce a formula for the sum of the squares of the positive divisors of $a$. -The section we ...
2answers
16k views

### Value of: $\frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + … + \frac {1} {\sqrt {80} + \sqrt {81}}$ [closed]

I have the following question: Find the value of: $$\frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + ... + \frac {1} {\sqrt {80} + \sqrt {81}}$$ The book ...
0answers
338 views

### Yet an other conjecture about odd numbers $n=a+b$ such that $a^2+b^2$ is prime

This question is related to A conjecture about an unlimited path and Any odd number is of form $a+b$ where $a^2+b^2$ is prime but I present it on its own if anyone would like to help finding ...
1answer
232 views

### A conjecture about an unlimited path

Conjecture: For all primes of the form $a^2+b^2$, there are natural numbers $s,t,u,v$ such that $\quad s^2+t^2,u^2+v^2$ are primes $\quad a+b=s+t$ $\quad u+v=s+t+2$ $\quad |s-u|+|t-v|=2$ This ...
2answers
553 views

### Finding all integers such that $a^2+4b^2 , 4a^2+b^2$ are both perfect squares

Are there any integers $a,b$, such that: $$a^2+4b^2 , 4a^2+b^2$$ are both perfect squares?
1answer
520 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 ...
11answers
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 ...
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 ...
2answers
100 views