Question

I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but according to Goldston

http://www.math.sjsu.edu/~goldston/twinprimes.pdf

it was stated nowhere until de Polignac. I'm hoping to resolve this issue by accessing either primary historical documents, or other reputable secondary sources (Goldston being one such example). I have looked at de Polignac's work, and he does indeed make a conjecture, but have been unable to find anything definitive (besides Goldston's statements) that there was no conjecture earlier. If this is too specific for MO, I'll remove the question. Thank you.

Answer 1

I don't have it to hand right at this moment, but Narkiewicz' The Development of Prime Number Theory is excellent on just this kind of question. It is a historiomathematical survey of prime number theory up to 1910, and also has discussions of later developments directly related to work done before 1910. It is historical, with very many references, and mathematical, in that it sketches many old proofs. It even has exercises.

Answer 2

Euclid never made a conjecture about the infinitude of twin primes.

It is possible to guess that he was making a conjecture on the basis of his text but it requires wishful thinking.

Here is the paper where de Polignac makes his general conjecture (which if true also implies the twin prime conjecture).

Regarding the NOVA show, Goldston makes a comment to those behind the NOVA segment (with a response) here:

http://discussions.pbs.org/viewtopic.pbs?t=45116

Answer 3

Hardy & Littlewood give what I believe to be the first quantitative version of the twin prime conjecture (actually, generalized to all even differences) as Conjecture B in their famous "Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes" on page 42. In particular, they conjecture that the twin-prime counting function $P_2(n)$ is

$P_2(n)=2C_2\frac{2}{(\log n)^2}(1+o(1))$ where $C_2=\prod\left(1-\frac{1}{(\varpi-1)^2}\right)$ with $\varpi$ running over the odd primes.