A June 8 Purdue University news release reports a proof of the Riemann
Hypothesis by L. de Branges. However, both the 23-page preprint
(from 2003) cited in the original release and a 124-page preprint
(from 2004) cited in a back-dated modified release seem to lack an actual proof.
Furthermore, a counterexample to de Branges’s approach by Conrey and
Li has been known since 1998. The media coverage therefore appears
to be much ado about nothing.

A May 26 preprint by Vanderbilt University mathematician R. F. Arenstorf
appears to come close to settling the long-standing question of the
infinitude of twin primes. While a hole has recently been found in the
proof, mathematicians remain hopeful that the proof can be corrected.

Josh Findley, a participant in the Great Internet Mersenne Prime Search (GIMPS),
identified the 41st known Mersenne Prime on the morning of
May 15. The discovery was confirmed on May 29, making 2^24036583 - 1
the largest known Mersenne prime, as well as the largest prime number known.