Sum of Squares and Cubes
(Jit here). There is a famous theorem of Lagrange that says every natural number $n$ is a sum of four non-negative squares (so that $0$ is allowed). Let's try to prove this theorem.
Posts
The Mysterious Tetrahedral Squares - An Adventure in Number Theory
Hey! Choo Ray here. I've been involved in giving sessions for the SIMO National Team and other trainings recently, so naturally I have to hunt for suitable problems. I feel that I've been looking at more problems recently than I have in the lead-up to my participations in IMO/other competitions! Of course, I spend less time on each problem, as my objectives are geared towards discovering problems with good ideas and instructive value rather than solving them myself. However, sometimes I find myself being led down a long rabbit-hole of theory that I apparently ought to know about. In this post I'd like to share about one of these experiences. One day, I was browsing contest collections on Art Of Problem Solving (AoPS). A question from the 2020 Bulgarian National Olympiad caught my eye: P4. Are there positive integers $m>4$ and $n$, such that a) ${m \choose 3}=n^2$ b) ${m \choose 4}=n^2+9$ I clicked on the link, thinking that it seems a rather routine proble...