Yes factoring is an active area. I think there are ranges (maybe $80$ digits?) where factoring is reasonably tractable but many integers have never been examined.

Hueristic tests tell us with extremely high confidence if much larger integers are prime. An actual proof certificate by ECM is never a problem in practice.

Some people care about numbers like $n!+1$ skipping many lower numbers. A large integer needs something interesting to make it desirable to crack

So I'll take your question as

What size are the smallest composite integers that people are trying to factor?

.

I'd guess around $200$ digits. I base this on the venerable and ongoing Cunningham project which started before 1925, and is still going strong. It is concerned with *Factorizations of $b^n \pm 1$, $b = 2, 3, 5, 6, 7, 10, 11, 12$, up to High Powers* Here are the $10$ **most** wanted composites. Also on that page are the slightly *less* desirable $24$ **more** wanted composites.

Most seem to be in $200$ to $300$ digit range.

The number one most wanted is $2,1207$- $C337$ This means that

$$2^{1207}-1=(2^{17}-1)(2^{71}-1)C337$$

The first two factors (of which the first is prime and the second a product of three primes) are explained by algebra and the fact that $1207=17\cdot71.$ The final factor has 337 digits and is know to be composite , but that is it (other than probably no easy factors).

That most wanted list is from $2016.$ Among the more wanted is a $278$ digit composite factor of $7^{359}+1.$ On the New factors received from July 3, 2017 to January 30, 2018O page one finds that a $70$ digit prime was extracted leaving a $208$ digit composite.

7, 359+ c278 2411303103482828219339285233829803927599026677829066502451625793020443. c208 NFS@Home snfs

This was a result from the distributed NFS@Home project using the special number field sieve.

Finally, on the new factorizations since January 30 2018 page one finds that this composite is the product of two primes. One listed and the other deducible from the given information.

7, 359+ c208 19792929490897848580163051528461396921535266011583367339063480695499122468294419091343065854809576593. p108 NFS@Home snfs

Browse those pages to see who decides what is wanted and why and much more.

knowledge, we might wonder whether every even number, and every multiple of 5, count as numbers that areknownnot to be prime, even if they are numbers that no one has ever happened to write down. If we do count those as known, what about multiples of 7, or 137? There will definitely be some clear cases, but I suspect that at most points in time, there will be some medium-largish number that no one has ever bothered to test for primality, that would take only a moderately long time to test (say, a few hours). Of course, that doesn't address your main question $\endgroup$ – Kenny Easwaran Oct 21 '09 at 5:42