about 12 years ago I came up with a pretty clever way to factor numbers that I never pursued the computational complexity of.

The basic algorithm is that you take some candidate X (which will be our 2048 bit number here) and classify your question (primality, whether it is the product of 2 primes, etc) --- once you have your question, Q, then you can pick a number Y0 to get X % Y0 = Z0 ... sometimes ~sqrt(X) works well, other times it's the closest prime factorial, etc.

now using those results, [Q, Y0, Z0], you can optimally pick Y1 and do the operation again, X % Y1 = Y2 ...

Like the Chinese remainder theorem each Z gives you information on the next optimum Y given your question Q ...

I called it tunnel factoring and saw some great early results ... but for some reason I haven't ever pursued it