Christian Goldbach (1690 – 1764) was a German mathematician famous for his eponymous Conjecture. Goldbach’s Conjecture is one of the most infamous problems in mathematics, and states that every even integer number greater than 2 can be expressed as the sum of two prime numbers. For example, 4=2+2, 6=3+3, and 8=3+5. While there have not been any counter-examples found up through 4 x 1018 (as of 2012), the conjecture has not yet been formally proven.
One of Goldbach’s earlier conjectures was that every odd composite integer could be expressed as twice a perfect square plus a prime. For example, 9 = 2(12)+7, and 15 = 2(22)+7. This week’s GeekDad Puzzle of the Week is simple: what are the two smallest counter-examples that disprove this conjecture?