# Twin Prime Conjecture Reference

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 I'm hoping to resolve this issue by accessing either primary historical documents, or other reputable secondary sources (Goldston being one such example).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.

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.

• Before there was Narkiewicz, there was Dickson. In his History, he mentions de Polignac, and doesn't mention anyone earlier. May 13, 2010 at 3:06

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

No one really knows if Euclid made the twin prime conjecture. He does have a proof that there are infinitely many primes, and he or other Greeks could easily have thought of this problem, but the first published statement seems to be due to de Polignac in 1849. Strangely enough, the Goldbach conjecture that every even number is a sum of two primes seems less natural but was conjectured about 100 years before this.

• Thanks for the link. It's good to see he responded to that, I too was a bit surprised they didn't mention Y or P in the song. Dec 3, 2009 at 14:33
• The PBS link seems to have gone away. Feb 14, 2014 at 21:46
• @GerryMyerson: I've added the relevant text via the Internet Archive. Feb 18, 2014 at 14:44

Hardy & Littlewood give (in 1923) 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

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