If Prime Numbers Become Increasingly Rare, Then Why Do They Keep Showing Up In Pairs?

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An impossible problem arrives by email

A strange email lands at Annals of Mathematics on April 17, 2013. It claims to contain a 50 page proof about one of the oldest unsolved problems in mathematics, and it comes not from a star professor but from an unknown mathematician who had once spent years working at Subway. The editors expect the usual story. As one mathematician says, the Annals gets a proof of the Riemann Hypothesis every other day, so the assumption is simple: send it to a referee, find the mistake in an afternoon, move on. But that does not happen. The referee keeps reading, looking for the corner where the carpet will bunch up, the fragile spot where a proof like this normally fails. Instead, each dangerous corner has been cut exactly right. By the end of the week, they realize: "Oh, damn, he did it." To understand why this was such a shock, Derek backs up to the actual problem: the twin prime conjecture. Twin primes are pairs like 11 and 13 or 17 and 19, primes separated by just one number. That sounds innocent until you remember the larger pattern. Primes get rarer as numbers grow. The average gap between primes near a number N is roughly ln(N). Around 100, the average gap is about 4.6. Around 1000, it is 6.9. Since the logarithm grows forever, the average spacing also grows forever. That makes twin primes feel like they should eventually die out. And yet they do not seem to. After a million, you quickly find 1,000,037 and 1,000,039. Past a billion, there is 1,000,000,007 and 1,000,000,009. Mathematicians have even found a twin prime pair of the form 2,996,863,034,895 times 2 to the 1,290,000 plus or minus 1, each number 388,342 digits long, about 260 pages if printed in a book. So the curiosity gap opens wide: if primes thin out, why do these pairs keep resurfacing? Checking examples forever is impossible, so the real challenge is not spotting twin primes, but proving they never stop.