Determine whether there are infinitely many twin primes, which are pairs of prime numbers that differ by 2.
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The question of whether there are infinitely many twin primes is still an open problem in mathematics. It is called the Twin Prime Conjecture and it remains unsolved.
The conjecture states that there are infinitely many primes p such that p+2 is also prime, giving us an infinite number of twin primes. This has been an unsolved problem for centuries, and while there has been progress and many computer searches have found large sets of twin primes, a complete proof still eludes mathematicians.
The Twin Prime Conjecture is closely related to the more famous and older unsolved problem, the Goldbach Conjecture. The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. Twin primes can be seen as a special case of the Goldbach Conjecture, where one of the primes is always 2.
Despite the efforts of many mathematicians, including recently by an online collaborative project called the Polymath Project, a rigorous proof for the Twin Prime Conjecture is still missing. Therefore, at present, we do not know for certain whether there are infinitely many twin primes.
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