Explore OEIS A000043, the sequence of Mersenne exponents p for which 2^p − 1 is prime. We unpack the double-primality condition, why Mersenne primes matter in number theory, and the real-world effort behind GIMPS—the distributed search that hunts for ever-larger primes. We’ll revisit Euler’s famous missteps with p = 41 and p = 47, discuss the difficulty of primality testing at huge scales, and why the sequence is thought to be infinite even if we haven’t proven it yet. A compact map for curious minds to understand primes, computation, and the ongoing journey of mathematical discovery.
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