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
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.
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