mgroeninger says:

I am not one hundred percent sure of the identity of the '300 year old sum', but I would guess if it referring to the question of ‘Mersenne Primes’. Here is my logic (culled mostly from the internet and what little I can remember after a rough holiday… so errors may be large):

The article probably refers to the Manchester Mark 1 computer (http://www.computer50.org/mark1/MM1.html). The Mark 1 computer was designed in 1948-49 by a team led by Professor Sir Frederic Calland Williams (http://www.computer50.org/mark1/williams.html) and then handed over to the Manchester Mathematics department, so the names and the timing appear to be right. Newman and Turing, who had used a previous machine in the summer of 1948 to pursue Mersanne Primes because they believed they were a “suitable subject for computer usage in pure mathematics” (http://www.computer50.org/mark1/maths.html#newman), appear to have used the Mark 1 for the same purpose in 1949. Wikipedia (http://en.wikipedia.org/wiki/Manchester_Mark_I) claims “the first realistic program to be run on the Mark I was a test of Mersenne primes, run in early April 1949.”

Mersenne Primes are named for Marin Mersenne, who referenced them in his Cogitata Physica-Mathematica in 1644. That would put it at close to 300 years old (http://primes.utm.edu/mersenne/). This is kind of speculating, though, since the idea of Mersenne Primes is older than Mersenne himself. But it is plausible that whoever compiled the article did not full understand what they were writing about.

However, Newman and Turing would not have “solved” anything regarding the Mersenne Primes. They merely tested integers which the machine was capable of testing, but were greater than known results, to see if a Mersenne Prime existed. While they were greaking new ground by testing unknowns, they did not actually find one (some information online says they were only ten or eleven tests from finding one… but I can’t find any good documentation of that).

Interestingly (and completely as an aside) the most recent Mersenne prime (the forty third) was just found this month by the GIMPS project (http://www.mersenne.org/).

Vaughan says:

Thanks for that (and for all the accompanying research as well)!