Largest Known Twin Prime Pair
On January 15, 2007 two distributed computing projects, Twin Prime Search and PrimeGrid found the largest known twin primes, 2003663613 · 2195000 ± 1. The numbers have 58711 decimal digits. Their discoverer was Eric Vautier of France.
On August 6, 2009 those same two projects announced that a new record twin prime had been found. It is 65516468355 · 2333333 ± 1. The numbers have 100355 decimal digits.
On December 25, 2011 PrimeGrid announced that yet another record twin prime had been found. It is 3756801695685 · 2666669 ± 1. The numbers have 200700 decimal digits.
An empirical analysis of all prime pairs up to 4.35 · 1015 shows that if the number of such pairs less than x is f(x)·x/(log x)2 then f(x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity.
There are 808,675,888,577,436 twin prime pairs below 1018.
The limiting value of f(x) is conjectured to equal twice the twin prime constant (not to be confused with Brun's constant)
(sequence A114907 in OEIS) this conjecture would imply the twin prime conjecture, but remains unresolved.
The twin prime conjecture would give a better approximation, as with the prime counting function, by
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