APMO · 5. Diophantine equations and representations of integers
Solve and write proofs for integer equations: squares between squares, constructions of numbers with a given form, bounds from divisibility, residues against factorials and powers, and infinite descent.
Show that an expression is not a perfect square by trapping it between consecutive squares.
Construct integers whose sum and product have a prescribed form.
Turn divisibility into size bounds and finish with a complete finite check.
Choose moduli that rule out cases in equations with factorials and powers.
Write a correct infinite-descent proof.