What is the first Mersenne prime exponent over 1000?

The story behind the answer

Mersenne primes have the form 2^p − 1, where p must be prime, but not every prime p works. Above 1000, the prime exponents 1009 and 1069 give composite numbers, while 2^1279 − 1 is prime—making 1279 the first Mersenne prime exponent over 1000. Although 2203 also works, it comes later.

Context: Mersenne primes are rare and are efficiently checked with the Lucas–Lehmer test; 2^1279 − 1 was one of the earliest large primes verified by computer in the 1950s. Memory tip: “Over 1000, skip two primes (1009, 1069); the next one, 1279, hits.”