Sum of two squares in many ways

Studente Anonimo
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!

Risposte
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

Studente Anonimo
Studente Anonimo
What you say is right... but not very useful in this case. Let me just give a tiny hint.
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$^2$.


____
$^1$ 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.

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