In the sequence of all integers, there are arbitrary long runs with no primes.

Yes, indeed. The basic construction constriction starts with factorial. As we know, N! = 1*2*3*4*...*N is the product of all integers from 1 through N. By definition, N! is divisible by every integer not exceeding N. Also, in general, a common divisor of any two numbers a and b divides their sum as well.

Thus equipped, we may claim that

N! + 2 is divisible by 2
N! + 3 is divisible by 3
N! + 4 is divisible by 4
N! + 5 is divisible by 5
...
N! + N is divisible by N

It follows that (N-1) consecutive numbers (N! + 2), (N! + 3), ..., (N! + N) are all composite.

N! grows astronomically fast. 10! = 3628800. 70! is larger than the Googol, which is a 1 followed by 100 zeros. 100! is a number with 158 digits. So we are fortunate to have a short and handy notation for this number. Factorials are indeed used everywhere in Mathematics. Their appearance in the Binomial Theorem and Taylor series would alone justify a special notation for this product. The word "factorial" is in use from 1800 and the notation N! from 1808. Here is a problem that puts to test your understanding of this notation.

Problem

Find integer m,n,k such that m!n! = k!.

Solution

Remeber the recursive formula for factorials:

Well, it's all here. Choose m arbitrarily and let N = m!, n = N-1, and k = N. Written explicitly, m!*(m! - 1)! = (m!)!.

Reference

1. R.Honsberger, More Mathematical Morsels, MAA, New Math Library, 1991