Two numbers have survived a century of mathematical scrutiny.
In short: Despite searching through hundreds of quadrillions of integers, mathematicians have only found two examples of a specific prime property. This video explores the mystery of Wieferich primes and why they remain one of number theory's most elusive puzzles.
A prime number $p$ is a Wieferich prime if $2^{p-1} - 1$ is divisible by $p^2$. While Fermat's Little Theorem guarantees this value is divisible by $p$, the requirement of $p^2$ is exceptionally rare. Only two such primes are known: 1093, discovered by Waldemar Meissner in 1913, and 3511, found by Nicolaas Beeger in 1922.
The significance of these primes stems from Arthur Wieferich's 1909 thesis, which linked them to potential counterexamples of Fermat's Last Theorem. Although heuristic arguments suggest there are infinitely many Wieferich primes, computational searches have failed to find a third even after testing every prime up to $10^{17}$. They represent a profound gap between what we believe to be true and what we can actually prove.
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- Summary by Intellect · source: Only 2 Primes Have This Property. We Don't Know Why.
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