See discussion here:

https://www.askmehelpdesk.com/math-sciences/prime-numbers-400-digit-number-49928.html

I'm curious as to what course you're taking that asks this?

I have seen this same question asked at least a dozen times since I began on this site. I do not know what signficance it holds that makes it so 'in demand'.

Maybe we can finally put this one to bed! Because this question keeps coming up every year or so, I decided to so some research. What I found is that the 15th Merseinne prime is:


M_{15} = 2^{\small 1279} -1


and it is 386 digits long. It's rounded value is 1.041 x 10^385. (for reference, see: Integer Lists: Mersenne Primes (http://www.tsm-resources.com/alists/mers.html))

Multiply this by a 15 digit prime, and you'll have a 400-digit number. So how to find a 15-digit prime? Use a "prime checker," available here: Big Primes: large list of prime numbers (http://www.bigprimes.net/primality_test/)
and test odd numbers starting with 100000000000001 until you hit a prime. Doing this I find that 100000000000031 is prime.

So multiply these two numbers together, and you'll have a 400-digit number that is the product of 2 primes.