The RSA numbers serve as a set of large testcases for algorithms for finding large prime factors. The largest of these, RSA-2048, has never been factored. It is generally expected that it will never be factored with purely classical resources; i.e., medium-scale quantum computers will be required to find its two factors.

The numbers themselves and historical factorizations may be viewed here: https://en.wikipedia.org/wiki/RSA_numbers

This question has a potentially long timescale, so to attempt to minimizing discounting-related distortions, I will resolve to PROB at the end of the year.

(I intend for this question to serve as a proxy for "when will we have working, scalable, quantum computers", without needing to quibble over definitions too much.)

May 19, 12:59am: PROB appears to not be an option. At the end of 2022, this market resolves to the mode.