A monster with 22,338,618 digits --------------------------------
We already had a pleasant surprise from prime numbers, which seem to have "clearly defined preferences for the final digits of the prime numbers that follow them". Now a new planet has just been discovered in this hushed universe: the number 274,207,281 – 1 is prime. This is a Mersenne prime (that is, of the form 2*p* – 1, for which an efficient primality test is known). The discovery of this monster of 22,338,618 digits is due to Curtis Cooper and his team. This American mathematician, a professor at the University of Central Missouri, is no stranger to this, having previously tracked down three other Mersenne primes between 2005 and 2013. In fact, he installed the Great Internet Mersenne Prime Search (GIMPS) software on his university's computers, and it was this collaborative system that helped him with his four discoveries.
Twin primes: still full of surprises! -------------------------------------
In Tangente 153, we had left off with two important conjectures, both proved on the same day, 14 May 2013. On the one hand, a weak conjecture on twin primes ("there exists an integer N less than 70,000,000 and infinitely many prime numbers p such that the interval ]p, p + N] contains a prime number") was proved by Yitang Zhang, of the University of New Hampshire. On the other hand, the weak Goldbach conjecture ("every integer greater than 7 can be written as the sum of three prime numbers") was proved by Harald Helfgott, of the École normale supérieure de Paris. Two "weak" versions of famous conjectures, proved nonetheless — let's not deny ourselves the pleasure! Progress has been made since then: driven by 2006 Fields medalist Terence Tao, the collaborative team of the Polymath 8 project developed Zhang's techniques, simplified his proof, and lowered the value of N to 4,680; the young British mathematician James Maynard (universities of Montreal and Oxford) then brought it down further, to 600, in November 2013. When will we get N = 2? The twin prime conjecture would then be proved!
Small and large gaps: more progress --------------------------------------------