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(edited 1 month ago)

does anyone know how to solve exponential diophantine equations
like 4^x + 6^y = z^2 by a) using modular arithmetic and b) without using modular arithmetic?

For the modular one, do you know the basics so what are the remainders of square numbers mod 3, 4, 8 say?
Generally without mod, youd try something like difference of two squares, so try and transform a sum of squares into a form which you can discuss the factors.

(edited 4 months ago)

Reply 2

mqb2766

Original post by user142615920

i don’t know much about modular arithmetic at the moment apart from essentially what it is, but i can probably read up on it. thank you for the help.

When you say solve using mod arithmetic, often the key thing is the value of n. So the quadratic residues mod n are in
https://en.wikipedia.org/wiki/Quadratic_residue
and its fairly easy to show that
z^2 = 0,1 mod 3
z^2 = 0,1 mod 4
...
so for instance you cant have a square number which gives 2 mod 4. This rules out any solutions for y=1 for instance (can you show that).

So if y is even here, youre really talking about pythagorean triples ... So similar to Q4 on MOG this year, its worth understanding
https://math.stackexchange.com/questions/3386302/if-x2-y2-z2-then-xyz-equiv-0-pmod60-pythagorean-triples
Similarly MAT 2023 had a question like are there integer solutions of the ellipse
x^2 + 3y^2 = 3332
Both of these are worth thinking about in addition to what youve posted.

It worth doing something, and posting some thoughts, even if it doesnt lead to the right answer. Often mod arguments are fairly concise, but the choice of n etc hides a fair bit of understanding.