Olympiad Number Theory

Hello!

This is sort of a Tricki-style experiment, but not exactly...

Please/feel free to contribute any ("algebraic") heuristic/trick/strategy here, no matter what level, or how broad/specific.

Thank you!

Here are some (not necessarily good) examples, to get us started...

1. A super-specific strategy: $\sin(x-y)\sin(x+y) = \sin(x)^2 - \sin(y)^2$ can sometimes help in simplifying trigonometric expressions. (What other functions satisfy $f(x-y)f(x+y) = f(x)^2 - f(y)^2$? See Putnam 1963 B3 for a partial answer.)

2. Another specific strategy: it can sometimes help to work modulo $4$ in Diophantine equations with squares. (And a more general related strategy: sometimes it helps to take a Diophantine equation modulo a specific number $m$, where the choice of $m$ depends on the context. For example, modulo $7$ helps to deal with cubes, since $7 \equiv 1\pmod3$ means that $a^3$ only takes on $\frac{7-1}{3} = 2$ nonzero values modulo $7$. This generalizes to mod $p$ to deal with $d$th powers, when $p\equiv1\pmod{d}$.)

3. A broader/more general strategy: deal with inverses by undoing inverse (radicals, logs, inverse trig, even basic stuff like $(x+1)-1$ to solve linear equations).

4. An algorithm for solving any system of polynomial equations: Buchberger's algorithm (thanks to Terry Tao for mentioning this, but unfortunately, as he said, I don't think we can teach this...)