We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 2¹⁰³⁹− 1. Although this factorization is orders of magnitude ‘easier’ than a factorization of a 1024-bit RSA modulus is believed to be, the methods we used to obtain our result shed new light on the feasibility of the latter computation.