What is the first Mersenne prime exponent over 1000?

The first Mersenne prime exponent over 1000 is 1279 because it is the smallest prime number that, when used as an exponent of 2, subtracted by 1, gives a prime number. Mersenne primes are of the form 2^n - 1, where n is a prime number. Starting from n=2, the sequence of Mersenne prime exponents results in prime numbers like 2, 3, 5, 7, 13, etc. In this case, 2^1279 - 1 is a prime number, making 1279 the first Mersenne prime exponent over 1000.