Twin Prime Conjecture: The Twin Prime Conjecture posits that there are infinitely many pairs of prime numbers differing by 2 (e.g., 3 and 5, 11 and 13). While primes become rarer as numbers grow, twin primes appear persistently. In 2013, Yitang Zhang made a breakthrough by proving there are infinitely many prime pairs with a gap of at most 70 million—later reduced to 246 through collaborative efforts. However, closing the gap to 2 remains out of reach. The conjecture is linked to the Hardy-Littlewood k-tuple conjecture and deep properties of the zeta function. Its resolution would illuminate the fine-scale distribution of primes and test the limits of analytic number theory. Though not a Millennium Problem, it is among the most celebrated open questions in mathematics, embodying the mystery of prime patterns.
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