Sum of two squares in many ways

[Studente Anonimo]

Hello everyone I have some questions for anyone interested.

(1) Find a positive integer which can be written as sum of two squares in at least three different ways.

(2) What is the smallest such integer?

(3) Prove or disprove: for every fixed integer [tex]k \geq 1[/tex] there exists a positive integer [tex]n_k[/tex] which can be written as sum of two squares in at least [tex]k[/tex] different ways.

I don't know the answers to (2) and (3).

Solutions in english please!

[perplesso1]

"Martino":(3) Prove or disprove: for every fixed integer $k≥1$ there exists a positive integer $n_k$ which can be written as sum of two squares in at least $k$ different ways.

It's true. Given $k$ distinct primitive Pythagorean triples $(a_1,b_1,h_1), ... , (a_k,b_k,h_k)$ the product $\prod_{i=1}^k (a_i^2+b_i^2) $ can be written in $k$ different ways, namely $(a_i \prod_{j \ne i}h_j)^2 + (b_i \prod_{j \ne i}h_j)^2$ for $1 <= i <= k$

[perplesso1]

"Martino":(2) What is the smallest such integer?

After analyzing all the sums $x^2+y^2$ with $1 <= x,y <= 50$ , my computer says

$325 = 1^2+18^2=6^2+17^2=10^2+15^2$

[perplesso1]

Using Martino's hint and interchanging $c^2$ with $d^2$ we can obtain another decomposition of $(a^2 +b^2)(c^2 + d^2)$ as sum of two squares. Using two Pythagorean triple $(a,b,k)$ and $(c,d,h)$ we can hope to find a third different decomposition, namely $(ha)^2 + (hb)^2$, and also a fourth one $(kc)^2+(kd)^2$. For example I found this

$65^2 = 16^2+63^2 = 25^2+20^2=33^2+56^2=39^2+52^2$

My english is very terrible...

[Studente Anonimo]

What you say is right... but not very useful in this case. Let me just give a tiny hint.

[tex](a^2+b^2)(c^2+d^2) = (ac+bd)^2+(ad-bc)^2[/tex].

PS. "I don't have" is fine. "I haven't got" is also fine, but it's either strictly british or less common (as far as I know).

[Zero87]

I don't have (or I haven't?$^1$) the solution of your problem yet, but I think I am next to the solution.
I post my observation in a spoiler way$^2$.

I think that this question is similar to the one of Gauss that a number $n$ can be written into a sum of different squares if and only if $n=l^2 m$ and $\forall p \le m$, $p|m$, $p -= 1 (mod 4)$ or $p=2$. $p$ is a prime number.

____
$^1$ English teacher of the first $2$ years of liceo said "I haven't", but the other teacher of the last three years said "I don't have"... I'm a little confused...
$^2$ My english is awful.