Here are the slides for a talk I am giving this weekend at the SERMON conference in Johnson City, TN. The basic question addressed is as follows.
Look at all numbers a such that a2+1 is prime. (The sequence starts 1, 2, 4, 6, 10... ) Goldbach conjectured that any number (other than 1) in that sequence is the sum of two previous numbers. So 2=1+1, 4=2+2, 6=2+4, 10=4+6, etc. We verify this for a up to 2*1014 and explore how easy it is to find these sums. This is joint work with Hester Graves.