Are there any known techniques for quickly finding a composite number with at least one large prime factor?
Being aware that the above might sound vague. Let's say quickly is anything less than O(n) complexity, and large factor as anything greater than sqrt(C) ,where C is the composite.
To be clear, I don't need to know what the prime factor is, instead I only need to know that the chosen composite contains a large prime factor.
I recall perusing properties of binomial coefficients with respect to the above, but I can't seem to find it.
The usual method is: find two large primes, and multiply them together. This has the great advantage that you are working with numbers approximately half the number of digits compared to your final composite number.
If you want to "happen upon" such a number rather than constructing it, you might try a large Mersenne number $M_p = 2^p -1, p$ prime, check that it's composite using Miller-Rabin, then trial divide by $q=2kp+1$ (hopefully finding nothing) until you are satisfied that the remaining factors are big enough.