The sesquicentennial of the prime number $2^{127} - 1$

Abstract

The year 2026 marks the 150th anniversary of a remarkable achievement in mathematics: in 1876, the French mathematician \'Edouard Lucas showed that the 39-digit number $M_{127} := 2^{127} -1$ is prime. This stood for 75 years as the largest known prime, and it remains the largest prime number discovered and certified without the aid of a mechanical device. We review the history behind this discovery and give a modern proof of the closely related Lucas-Lehmer test, which is still the main engine behind the certification of large prime numbers.