## Thursday 17 August 2017

### Erdös Aaron Numbers

A while back I was on about Erdös-Etc-Etc numbers which establish your status in mathematics depending on whether you danced with the Prince of Maths or danced with a man who danced with a girl who danced with the Prince of Maths [sing it]. Here's a nice Numberphile story by Carl Pomerance about Erdös numbers and the triggers of creativity. It starts in 1974 when baseball star Hank Aaron equalled [N=714] and then beat [N=715] the home-run record held by Babe Ruth since 1935. Like 1729 the Taxi-cab Number, certain numbers sing to mathematicians. Pomerance, a baseball fan, pondered on the numbers 714 and 715 which were being headlined all that Spring and noted that the prime factors of these two consecutive integers included all the primes up to 17 without dupes:
714 = 2 x 3 x 7 x 17
715 = 5 x 11 x 13
the sum of the prime factors is also [marginally] interesting:
2 + 3 + 7 + 17 = 29
5 + 11 + 13 = 29
Pomerance wrote a jokey-serious paper about these inter-weavings; which attracted the attention of Paul Erdös; who came down and started a fruitful collaboration with the young Pomerance; which kick-started the latter's career. Years later, Erdös and Aaron are being given honorary degrees at the same place and Pomerance is able to introduce them . . . and get them to sign the same baseball [preserved R]: giving Aaron an enviable Erdös Number of 1.  Sweet.

I must have an  Erdös Number. And in the nature of things it is going to be much less than Heinz 57. There are 500+ people with Erdös = 1, and very few mathematicians with E# higher than 8. Indeed it is almost a distinction to have a really high E# because that means you've been fossicking about on the most distant frontiers of maths. In this deeply databased age, it is, of course, the kind of thing that can be computerised and a number of large-hearted, time-rich people offer help in finding your E#.  You can also game the system if you're well positioned when the Apocalypse starts.