I'd put the factors of all outstanding RSA numbers in an arXiv paper, cc'd to my lawyer and several trusted friends, and announce the release of the algorithm on Github in a week. If the solution was a device, then blueprints/quantum circuits/etc.

What I'd do next -- and whether I'd release it -- depends on the public reaction. My first priority would be to light a fire under as many asses as possible, but without hurting anyone.