Twin Prime Conjecture

Determine whether there exist infinitely many twin primes, that is, infinitely many pairs of primes (p, p+2).

Background

In discussing Turing’s terminology of "circular" versus "circle-free" programs, the authors present an example program that outputs a new digit each time it finds a new twin-prime pair. They note that whether this program is circle-free depends on whether there are infinitely many twin primes.

They explicitly mark the status of the twin prime question as open, using it to illustrate that a program can fail to produce infinitely many output digits without being caught in a simple looping behavior.