By Fermat's little theorem, \(2^{p-1} - 1\) is divisible by any odd prime \(p\). A Wieferich prime is one where it's divisible by \(p^2\). Only two are known: 1093 (found by Wieferich's student Meissner in 1913) and 3511 (1922) — and there are no others below \(6.7 \times 10^{15}\).
They have a famous connection to Fermat's Last Theorem: Wieferich proved in 1909 that if the first case of FLT failed for an exponent \(p\), then \(p\) would have to be a Wieferich prime — a striking constraint decades before Wiles's proof. Whether infinitely many exist is unknown.