The exciting new results by Zhang and others about bounds on the gaps between pairs of primes have been getting a fair amount of press, which is great! Some of them have gotten me wondering about the origins and history of the Twin Prime Conjecture. My searches into this question have been so far been unsatisfying.
Several articles claim that the conjecture can be attributed to Euclid:
Some attribute the conjecture to the Greek mathematician Euclid of
Alexandria, which would make it one of the oldest open problems in
mathematics. (from here)
This isn't very satisfying. It's possible that this is true, but to my knowledge Euclid's extant works do not contain such a conjecture, or even conjectures at all. (overview) So if this is true, Euclid's claim to of the Twin Primes Conjecture must have come from later sources.
Wikipedia has only the following weak statement to offer:
The question of whether there exist infinitely many twin primes has
been one of the great open questions in number theory for many years. (from here)
My many Google searches have not been successful in getting better information. Can anyone share a trustworthy reference about when the Twin Prime Conjecture was first stated? Bonus points if it gives some of the further history of this conjecture. The Wikipedia article only picks up in the early twentieth century. Thanks!
Best Answer
Too long for a comment : I believe that its origins will be forever lost in the mist of time,
for the following very simple reason :
Euler was already aware in the eighteenth century that all primes except for $2$ and $3$ are of
the form $6n\pm1.~($At any rate, such a trivial statement is relatively easy to either discover
or understand, even by people with only the most basic mathematical knowledge$).~$
Then the next question which naturally arises is about the density of those “lucky” values of
n for which both neighbors of $6n$ are simultaneously prime. So basically all that's left to do,
after first dispensing with certain formalities pertaining to what is considered to be academi–
cally acceptable mathematical etiquette, such as actually proving that their number is indeed
infinite $($most likely by using some painfully obvious argument based, say, on reduction to
the absurd, and the like : as in the case of proving that there are an infinite number of primes,
for instance$),~$ would be getting down to the really hard part of actually quantifying their
frequency, and then venturing to offer a mathematical explanation for the experimentally
obtained results $\ldots$
Except that —oh, wait a second— remember that first “easy” half we were talking about just
earlier ? Well, as “luck” would have it, it turned out to be not so easy after all $\ldots$ So that's it,
in a nutshell.