Prime numbers—the indivisible darlings of mathematics. Just when you thought they couldn't get any more intriguing, they pair up to form what we affectionately call "twin primes." These are pairs of prime numbers that differ by exactly 2, like the inseparable (3, 5), (5, 7), or the dynamic duo (17, 19). The big question that's been tickling mathematicians' brains for centuries is: Are there infinitely many of these twin primes, or do they eventually run out, leaving us with a lonely prime singleton? The Twin Prime Conjecture: A Never-Ending Story? The twin prime conjecture posits that there are infinitely many such pairs. Despite the simplicity of its statement, proving (or disproving) this conjecture has been as elusive as finding a needle in an infinitely expanding haystack. The conjecture was first articulated in 1846 by Alphonse de Polignac , who suggested a broader claim: for any even number $2k$ , there are infinitely many prime pairs $(p,p+2k)$ . The cas...
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