Articles of sums of squares

Clarification on how to prove polynomial representations exist for infinite series

With reference to this question, I would like a clarification of the comment given by @Ant (but someone else could answer instead). I basically have 2 questions: Is there any formal way to prove that there exists a polynomial representation for the sum of the first $n$ natural numbers to the $m^{th}$ power ($1^m+2^m+3^m+\cdots+n^m$) without […]

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?

When is the sum of two squares the sum of two cubes

When does $a^2+b^2 = c^3 +d^3$ for all integer values $(a, b, c, d) \ge 0$. I believe this only happens when: $a^2 = c^3 = e^6$ and $b^2 = d^3 = f^6$. With the following exception: $1^3+2^3 = 3^2 + 0^2$ Would that statement be correct? Is there a general formula for when this […]

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I’m working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we find all sequences of non negative integers ($\forall i\in \mathbb{N} \,\, a_i\in \mathbb{N}$) such that $$\begin{align} \forall i,j,k,l\in\mathbb{N} && i^2+j^2=k^2+l^2 &\Rightarrow a_i^2+a_j^2=a_k^2+a_l^2\end{align}$$ My try: I know […]

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$. I feel that the problem basically uses algebraic manipulation even though it’s in a Number Theory textbook. I don’t realize how to show $(a^2+b^2+c^2)^2$ […]

Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$

Problem Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$ My attempt was, Since $p$ divides $n^4 + 1 \implies n^4 + 1 \equiv 0 \pmod{p} \Leftrightarrow n^4 \equiv -1 \pmod{p}$. It follows that $(n^2)^2 \equiv -1 \pmod{p}$, which implies $-1$ […]

Number writable as sum of cubes in $9$ “consecutive” ways

Let’s say that a given $n\in\mathbb{N}$ is writable as sum of cubes in $k$ consecutive ways if it can be written as sum of $j,j+1,\ldots, j+(k-1)$ nonzero cubes, for some $j\geqslant 1$. For example, $26$ is writable as sum of squares in $4$ consecutive ways, for \begin{align} 26 &= 1+25 &&\text{($2$ squares)}\\ 26 &= 1+9+16 […]

Primes as sum of squares in finite field

Is there an algorithm to find integers such that the following holds true : ${a_1}^2 + {a_2}^2 \dots + {a_k}^2 \equiv p $ $(mod $ $n)$, where $p$ is a prime and $n$ is any general integer. By Lagrange 4 square theorem it is known that value of $k$ will be atmost 4. Also according […]

For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?

You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of which is the sum of two squares, is itself a sum of two squares. That is to say, the set of all sums of […]

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who’s Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the series results in: $3^2 + 4^2 = 5^2$. The second results in: $10^2 + 11^2 + 12^2 = 13^2 + 14^2$. The thrid: […]