There was once a man called John Titor, who turned up on bulletin boards claiming to be a time traveller. He presented elaborate documentation to back his case, including photographs of his time machine, his uniform, detailed operations manual and tools for maintaining the machine. He even showed light bending around the singularity used to power the machine. He even gave a convincing explanation of how the machine worked. He fooled a lot of people, but most people never heard of him and most of those who did believed it was a hoax.

Convincing a bunch of people you can factor large numbers isn't that hard. And there have been many claimants. But there are security researchers out there that publish lists of numbers to factor of various sizes, including ones (only) they know the factors of, and ones no one knows the factors of.

However, consider what happens if you convince a small number of people you really did it. Most people will either disbelieve it until their bank accounts are emptied, or they will believe it to be no more important than any other security breach.

There are three main groups of people who would understand what it meant. The first are mathematicians who would want to understand the factoring method to push forward their research. The second are Governments and large corporations who depend on factoring for security. The third group are hackers who would want to use it to steal secrets and money.

The second and third group probably wouldn't have your best interests at heart. And many of the first group are working for the second group. So pretty much you should boast about it on conspiracy forums and the like and tell people who don't necessarily understand your accomplishment. At least you'll get a Wikipedia page and people will remember you after you disappear.