# Solving Diophantine equations in math competitions

what are the common methods/techniques in BMO/SMC when it comes to solving diophantine equations?
Original post by yi123456
what are the common methods/techniques in BMO/SMC when it comes to solving diophantine equations?

It really depends on the type of equation(s). Do you have any of books/notes which cover the basics or any example questions youre stuck with?

Edit - in the brief preparation sheet
https://archive.ukmt.org.uk/docs/BMO%20Preparation%20Sheet.pdf
At BMO level, most questions in this area will involve finding integer solutions to equations (i.e. Diophantine Equations) for which an understanding of factorisation and the significance of primes is essential. A common situation is to find integer solutions of, say, xy + x + y = 2004 or similar which relies on realising that xy + x + y + 1 can be factorised to (x + 1)(y + 1). Some knowledge of modular arithmetic may be particularly helpful at BMO2 level. BMO1 often uses (implicitly) arithmetic modulo 10, so the idea of extending this is quite important. Number bases, rules for divisibility and the idea of parity are all helpful. Fermat's Little Theorem could be useful for BMO2.
(edited 1 month ago)
Maybe Fermat-Euler Theorem, Wilson's?