Twin Primes and Other Conjectures

Number Theory · Axiom Academy

LESSON Twin Primes and Other Conjectures Exploring the deepest unsolved mysteries about prime numbers The Twin Prime Conjecture states that there are infinitely many pairs of primes that differ by exactly 2. These pairs are called twin primes. (3, 5) - the smallest twin prime pair As numbers get larger, twin primes become increasingly rare, yet we believe they never stop appearing. Despite centuries of effort, no one has proven this conjecture. While we don't know if there are infinitely many twin primes, Viggo Brun proved in 1919 that the sum of the reciprocals of all twin primes converges to a finite value, now called Brun's constant . This is remarkable because the sum of reciprocals of all primes diverges (grows to infinity). This suggests that twin primes, even if infinite in number, are much "sparser" than all primes. B = 1/3 + 1/5 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + ... Current estimate: B ≈ 1.902160583104... Computed using twin primes up to very large values 3. Zhang's Breakthrough on Bounded Gaps In 2013, Yitang Zhang made a stunning breakthrough by proving that there are infinitely many pairs of primes that differ by at most 70,000,000. This was the first time anyone had proven that prime gaps are bounded infinitely often! Following Zhang's work, the Polymath project improved this bound dramatically: Zhang (2013): gap ≤ 70,000,000 Polymath (2014): gap ≤ 246 (unconditionally) Assuming certain conjectures: gap ≤ 6

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