Twin Prime Conjecture
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The twin prime conjecture is a yet unproven conjecture that states that there are infinitely many pairs of twin primes. Twin primes are primes of the form 
and 
.
Possible Proofs
Using an infinite series
One proof that there are infinitely many twin primes involves showing that the sum of the reciprocals of twin primes diverges. A strategy to prove that there are infinitely many twin primes is to consider the sum of reciprocals of all the twin primes:
Unfortunately, it has been shown that this sum converges to a constant B, known as Brun's constant.
This could mean either that there are finitely many twin prime pairs or that they are spaced "too far apart" for that series to diverge.
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