Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Tagged diophantine-equations — 4 entries. Clear
Ramanujan guessed in 1913 that x² + 7 = 2ⁿ has exactly five solutions; Nagell proved him right in 1948 via the arithmetic of Q(√−7).
Are 8 and 9 the only consecutive perfect powers? Catalan asked in 1844; Mihăilescu proved it in 2002 with a short cyclotomic argument.
a + b = c: how large can the radical of abc be relative to c? A statement about the deep tension between addition and multiplication — and home of mathematics' most controversial claimed proof.
No positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. Three hundred fifty-eight years, ten million pages of attempted proofs, one margin note too long to contain.
