Zaremba's conjecture, from 1971: for every positive integer q, there exists a coprime a such that all partial quotients in the continued fraction expansion of a/q are bounded by some absolute constant. The conjecture has resisted proof for over fifty years.
Shkredov proves it for prime denominators. For sufficiently large primes q, there exists a coprime to q with all partial quotients bounded by O(√(log q)). The bound is not the conjectured absolute constant — it grows, slowly — but the existence proof for primes is a major structural advance.
The result also comes with quantitative estimates: the number of such well-behaved a values is bounded below, and there are bounds on the count of a where the sum of partial quotients is at most O(log q · √(log log q)), improving earlier results of Korobov and Larcher.
The continued fraction expansion of a/q encodes the Euclidean algorithm applied to (a, q). Bounding the partial quotients means the algorithm converges uniformly fast — no single step takes a disproportionate fraction of the work. Zaremba's conjecture says this uniform convergence is always achievable if you choose a correctly.
The restriction to primes is not incidental. Prime denominators have the simplest multiplicative structure, and the techniques exploit the additive combinatorics of Z/qZ in ways that require q to be prime. The general case — all q, not just primes — remains open.
Fifty-five years for the prime case. The general case waits.