A Prime-2 Gap Challenges a Proof About Catalan’s Constant
Exact arithmetic exposes a prime-number accounting gap in a claimed proof about Catalan’s constant, while leaving its irrationality open.
number theory
The Prime-Counting Matrix That Caught Two-Thirds of Zeta’s Zeros
An unconditional theorem puts at least two-thirds of zeta’s nontrivial zeros on the critical line—without resolving the Riemann hypothesis.
A Prime-by-Prime Test Solves a Surprisingly Global Matching Puzzle
A divisor-matching puzzle has a surprisingly local solution: prime-by-prime counts guarantee a global coprime pairing.
The Zero-Sum Mystery: Why Two Piles Set the Toughest Benchmark
A 1992 conjecture asks how few zero-sum groups can hide in a sequence. A new proof shows two piles are the hardest arrangement to beat.
A New Counting Trick Catches Rare Spikes in the Zeta Function
A new method counts unusually large values of the Riemann zeta function, nudging a long-standing sixth-moment bound in a better direction.