Is there any written work by Euclid about consecutive primes differing by two? What work was done on the problem from the time of Euclid about 2,300 years ago to the time of Polignac in 1849?


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I am not sure how the association of Euclid with twin primes got started, but even Brittanica writes (although not, to its credit, Wikipedia):"Greek mathematician Euclid (flourished c. 300 bce) gave the oldest known proof that there exist an infinite number of primes, and he conjectured that there are an infinite number of twin primes". The first part does occur in Elements IX.20, and IX.36 further describes how to produce perfect numbers from "Mersenne primes", but the ending is a fable. Or as Jason Dyer charmingly put it on MO:"It is possible to guess that he was making a conjecture on the basis of his text but it requires wishful thinking". The whole Twin Prime Conjecture Reference thread there is instructive.

Apparently, Dickson's History of the Theory of Numbers mentions de Polignac's Six Arithmetical Propositions Deduced from the Sieve of Eratosthenes (1849) as the earliest source conjecturing the infinitude of twin primes. But even then according to that same Britannica:

"Very little progress was made on this conjecture until 1919, when Norwegian mathematician Viggo Brun showed that the sum of the reciprocals of the twin primes converges to a sum, now known as Brun’s constant... The next big breakthrough occurred in 2003, when American mathematician Daniel Goldston and Turkish mathematician Cem Yildirim published a paper, “Small Gaps Between Primes”, that established the existence of an infinite number of prime pairs within a small difference (16, with certain other assumptions)".


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