Stanley Skewes Proved a Number Existed Without Anyone Being Able to Reach It

Carl Friedrich Gauss noticed in his teens that the count of primes up to any number x is very close to the logarithmic integral li(x). Plot the two side by side from 2 up to anything you have a computer for, and the smooth curve sits just above the staircase: π(x) < li(x), every time, with the gap growing. Gauss conjectured the inequality held forever.

In 1914, John Edensor Littlewood proved Gauss wrong. The two curves cross. In fact, they cross infinitely often — π(x) eventually overtakes li(x), then loses the lead, then takes it again, and the trade keeps happening as x grows. What Littlewood did not do was tell anyone where the first crossing lies. His argument was an existence proof; it conjured the crossover without locating it.

Nineteen years later, Stanley Skewes, working under Hardy at Cambridge, did the locating. Sort of. Assuming the Riemann hypothesis, Skewes proved the first crossing happens before e^(e^(e^79)), which is roughly 10^(10^(10^34)). In 1955 he removed the Riemann assumption and got 10^(10^(10^964)). For years it sat in textbooks as a kind of reductio: a number you could write down but not really write down.

Since then the bound has collapsed. Computer-aided work using zeros of the Riemann zeta function has pushed the estimated crossover to about 1.4 × 10^316 — still beyond any conceivable enumeration, but a perfectly modest 316-digit number. The point of Skewes's number was never the number. It was that a true statement about the integers could leave no trace anywhere a human would think to look.