Wow -- I'm impressed. I just threw that out there to show how hard factoring is -- I didn't think anyone would actually do it. Ummm... Do you work for the NSA?
I don't work for the NSA—I just have a laptop, and a copy of Mathematica. The factorization took about 10 seconds.
Mathematica is really a pretty remarkable piece of software. I doubt it's world class for factorization, but it gives you easy access to a lot of pretty darn good algorithms for a wide range of problems.
There's 1 sentence on the implementation of FactorInteger in the docs, FWIW:
OK, so take a 64 bit integer. That's the typical size of an integer in a modern programming language. A signed integer can represent 2^63-1 which is 9223372036854775807.
Imagine a "big integer" which is a composite of multiple 64-bit integers. With a big integer made up of two regular integers, you can represent the value 170141183460469231731687303715884105727. With three, you get to 3138550867693340381917894711603833208051177722232017256447 if I'm doing the math right. That's already in the ballpark of the number you mentioned - just three regular integers together.
Granted, I'm not talking about how difficult the factoring is, but it's worth noting that the size of the numbers are pretty small to a computer. Modern computers are really fast and can brute force a large number of computations. Mathematica probably has a fairly optimized general purpose factoring algorithm as well.
A number like this might be more difficult to factor:
I would say factorisation is more tedious than hard. Modern computers can perform several billion operations per second. The simplest algorithms take O(N) time, that is to factorise N, you need to perform around N calculations. There are better algorithms which run in logarithmic time, so you only need log(N) calculations (and some which are better than that). Even with what you and I think of as big numbers, this is fast.
You get problems with numbers over 100 digits. I recently entered a challenge where one of the questions involved prime factoring a 130 digit number. That was a record feat a couple of decades ago, now it takes a day or two on a modern computer.
Anyone care to elaborate on the downvote? Given that the parent seemed surprised that it was possible to factorise a 'big' number quickly on a modern PC, I thought I'd elaborate.
Brute force search is O(N), the sieve of Eratosthenes is O(N log(log(N))). The Quadratic sieve is good up to 100 digits and runs in O(exp(sqrt(log n log log n))). Some utilities you can use for this are YAFU, MSIEVE or GGNFS.
