Number theory offers a host of problems that are remarkably easy to state but fiendishly difficult to solve. Many of these questions and conjectures feature prime numbers–integers evenly divisible only by themselves and 1.
For instance, primes often occur as pairs of consecutive odd integers: 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. So-called twin primes are scattered throughout the list of all prime numbers. There are 16 twin prime pairs among the first 50 primes. The largest known twin prime is the 32,220-digit pair 318032361 x 2107001 +/–1, found recently by David Underbakke and Phil Carmody.