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 exponential-diophantine — 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.
Gaps between perfect powers grow without bound. Catalan settled gap 1 (8 and 9); every other fixed gap is open, and abc would imply them all.
\(A^x\) + \(B^y\) = \(C^z\) with all exponents greater than 2 forces a common factor among the bases. Fermat's Last Theorem's living descendant, with a million-dollar bounty.
