Integer solutions to $x^2 + (x-y)^2 = 17^2$, with $x\ge0$

Question

I need help finding integer solutions to $$x^2 + (x-y)^2 = 17^2$$ with $x\ge0$

I know that this is of the form of the pythagorean theorem but I am quite unsure about how to proceed in finding all integer solutions for it.

This question was given to me by my teacher as a challenge question in our precalculus course.

https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple — lab bhattacharjee, Dec 18 '22 at 07:03
Make Pythagorean triplet choices: $ m^2+n^2=x, m^2-n^2=17, 2 m n=x-y$ — Narasimham, Dec 18 '22 at 19:05

score 1 · Accepted Answer · answered Dec 18 '22 at 05:26

1

Your question is solved once you find all integer solutions to $$a^2+b^2=17^2$$ This equation of course admits a solution as $17^2=17\times 17$ and hence the only primes dividing it are of the form $4k+1$ (if this statement doesn't make sense, you should check out Fermat's Two Square Theorem or just check that one solution is $(15,8)$).

Once you have one solution, you can find all the solutions using this method.

answered Dec 18 '22 at 05:26

Sayan Dutta

8,831

thank you for your help. The link was helpful, but once I derive an equation for a line that passes through a known value and some $(x, y)$, what is the next step to finding the solutions? The link doesn't quite provide clarity on this. – Jared Dec 18 '22 at 07:20
@Jared the link provided has got a few more links. Did you try them (especially the book or the blog post) to see whether things are clearer? Since you have accepted the answer, I guess, it's clear now. If not, please tell me and I'll try editing the answer to provide more details. – Sayan Dutta Dec 18 '22 at 07:54

Integer solutions to $x^2 + (x-y)^2 = 17^2$, with $x\ge0$

1 Answers1