Happy 2021 everyone! As a way to start out the new year, what’s better than an article about the mathematical properties of the number 2021 to prepare for the next Mathematical Team Competitions? To start out, 2021 is not a prime number, its factorization is: Hence, be prepared for a lot of chinese remainder theorem […]

# Category: Number Theory

## IMO 2019/4

Find all couples of integers which satisfy The Idea We notice that work and we claim that other solutions don’t exist, so we want to get an upper and lower bound for . Proof Now let be the greatest exponent of prime in the scomposition of , we notice that and . Hence, Now we […]

## A Strange Recursion

This is a really nice recursion I wanted to share. We have to find the general formula for the nth term of the succession. We have: Also: By subtracting the two expressions and simplifying we get: Now, since we can further simplify to get the final form: By solving the recursion’s equation we can find […]

## Nice Post-Dinner Diophantine

Today after dinner, I really wanted to do a geometry problem but this diophantine equation just suddenly appeared in front of my eyes and, because it’s been a while since the last number theory problem, I decided to give it a shot. Prove that has no integer solutions. First we notice that the former equation […]