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Primes, those numbers (as some middle school teacher probably tried to make you memorize some years ago) whose only factors are themselves and one (2, 3, 5, 7, 11 etc.), recently made the news.  This is usually a big deal, because the low hanging fruit in the number theory of primes, like most mathematical objects under study for at least a couple thousand years, was consumed long ago.

It’s been known since at least Euclid that there are infinitely many primes (for an accessible proof why, see here).  More tantalizing has been the conjecture that there are infinitely many so-called twin primes—consecutive prime numbers spaced only two apart from one another (3 and 5, 5 and 7, 11 and 13, 17 and 19 etc.).  As is often the case with conjectures of this sort in number theory, computers continuously search for larger and larger candidate pairs satisfying this simple yet maddeningly elusive condition, for while every once in a while a new twin prime pair is found (the largest being the 18,000 digit long monster 3756801695685 · 2666669 ± 1), no one has yet proved that there should be infinitely many.
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