Breakthroughs
Significant results in mathematics, computer science and physics. The bar is deliberately high, so quiet weeks produce nothing at all rather than filler.
For the earlier hand-written round-ups, see 2026 so far and 2025.
Latest digest
Breakthroughs Weekly — 4 October 2026
Graham’s 1971 rearrangement conjecture is settled for all large primes
arXiv · 17 Feb 2026 · Combinatorics
Huy Tuan Pham and Lisa Sauermann proved that, for any prime p, a set of nonzero residues can always be listed in an order where every running total is different, a question Ronald Graham posed in 1971. Their paper handles sets up to a power-law fraction of the prime’s size, and combined with earlier work by Müyesser, Pokrovskiy, Kravitz, Bedert and colleagues, this settles the conjecture for all sufficiently large primes. The argument leans on randomness together with anti-concentration estimates and Fourier analysis. It was posted in February and is a preprint without a journal reference, but it reached wider attention on 28 September 2026 when Quanta Magazine profiled the full line of work, with Princeton’s Noga Alon among the experts commenting on it.
Earlier digests
2026
Breakthroughs Weekly — 20 September 2026
A 2004 conjecture on random regular graphs, unnoticed for nearly a year, draws expert praise once Quanta profiles the completed proof.
Breakthroughs Weekly — 14 September 2026
A near-linear-time algorithm delivers a new, more efficient proof of the four-color theorem, cutting coloring time from quadratic to O(n log n).
Breakthroughs Weekly — 8 September 2026
A finite-time blowup proof for forced 3D Euler flows earns Terence Tao's praise as a Navier-Stokes stepping stone, and a five-decade percolation conjecture falls.
Breakthroughs Weekly — 31 August 2026
Claude pushes a Riemann zeta bound to 67%, a 30-year discrepancy-theory bound falls, and detectors catch solar neutrinos scattering coherently.
Breakthroughs Weekly — 27 August 2026
A 1939 conjecture on polynomial maps falls above two dimensions, and reactors show a faint antineutrino afterglow long after shutdown.