Articles of sums of squares

How to prove Sum of squares of $n$ numbers is unique

I was solving a programming problem where I needed to prove that sum of squares of $n$ numbers is unique. i. e. if calculate the sum of squares of equal number of positive integers then if they are same it means that all numbers are the same. How do I prove it? The range of […]

sum of square root of primes 2

I dont know how to solve the problem below. (1) $p[1]$, $p[2]$, $\ldots$, $p[n]$ are distinct primes, where $n = 1,2,\ldots$ Let $a[n]$ be the sum of square root of those primes, that is, $a[n] = \sqrt{p[1]}+\ldots+\sqrt{p[n]}$. Show that there exists a polynomial with integer coefficients that has $a[n]$ as a solution. (2) Show that […]

Is there any formula to calculate the number of different Pythagorean triangle with a hypotenuse length $n$, using its prime decomposition?

Lets define $N(n)$ to be the number of different Pythagorean triangles with hypotenuse length equal to $n$. One would see that for prime number $p$, where $p=2$ or $p\equiv 3 \pmod 4$, $N(p)=0$ also $N(p^k)=0$. e.g. $N(2)=N(4)=N(8)=N(16)=0$ But for prime number $p$, where $p\equiv 1 \pmod 4$, $N(p)=1$ and $N(p^k)=k$. e.g. $N(5)=N(13)=N(17)=1$ and $N(25)=2$ and […]

Numbers which are not the sum of distinct squares

We are defining square factorization as representation a positive natural number as sum of squares of different positive, integer numbers. For example $5 = 1^2 +2^2$ and $5$ has no more representation. But one number can possess more representations, eg. $30$. $$30 = 1^2 + 2^2 + 5^2 = 1^2 +2^2 + 3^2 + 4^2$$ […]

Let $F$ be a field in which we have elements satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements satisfying $d^2+e^2 = −1$.

Let $F$ be a field in which we have elements $a, b$, and $c$ satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements $d$ and $e$ of $F$, satisfying $d^2+e^2 = −1$. Any hint? This is an excercise from the book: The Linear Algebra a Beginning Graduate Student Should Know; Golan

Are there identities which show that every odd square is the sum of three squares?

I am looking for algebraic identities of the form $$ (2n+1)^2 = f(n)^2 + g(n)^2 + h(n)^2, $$ where the functions are polynomials in $n$. EDIT: Evidently $(6k)^2 = 36k^2$ is trivially the sum of three squares when $k$ is odd. We also have the identity $$ (6k+3)^2 = (2(2k+1))^2 + (2(2k+1))^2 + (2k+1)^2. $$ […]

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 any algebraic restrictions on [i.e., relationships between] the elements which are well-known or easy to prove? I’m thinking of things like […]

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 […]