One important application is to the field of cryptography. Computational number theory . There is an online primality-verifier which uses the Elliptic Curve primality test. Likewise an online factorer . Other web sites with this focus One of the pages devoted to Mersenne primes etc.
Several techniques are available for factoring large integers quickly.
Detailed descriptions of the elliptic curve method . A free implementation of the Elliptic Curve method of factorization. Several large-scale factoring projects involving many machines and many people. Here is a call for participants , which resulted in a completed factorization. The factorization of RSA129 .