Yves Gallot's Proth.exe (Another of the Prime Pages' resources)
 In 1997, the "Proth program" `Proth.exe` was created by Yves Gallot for the search for large prime factors of Fermat numbers and implemented the following theorem: Proth's Theorem (1878): Let N = k*2n + 1 with k < 2n. If there is an integer a such that a(N-1)/2 = -1 (mod N), then N is prime. Now, the "Proth program" has been expanded to cover the primality test of all numbers N of the form k*bn + 1 or k*bn - 1. It is designed to allow test of any number of these forms and the largest of them, which can be tested on modern computers, have more than 10,000,000 digits! Then in practice, the difficulty of the test is quickly multiplying the large numbers involved. Proth is highly optimized for the test of large numbers (more than 10,000 digits). Discrete Weighted Transform and Fast Fourier Transform multiplication is used for squaring or multiplying, plus fast modular operations (using the special form of N) are also employed for speed purposes. Proth.exe has been used to find the most primes (of any program) on the list of Largest Known Primes. It has also been used to find most of the largest known non-Mersenne primes.