So, last night, at a coffee shop, I just sat down and the answer was so clear! I was looking at the case for n=5 but the general case was pretty obvious after understanding it for n=5

The theorem to prove: For every positive integer n, there is a sequence of n consecutive positive integers containing no primes.

I gave myself a hint:

Suppose n is a positive integer. Let x = (n + 1)! +2

Show that none of the numbersxx+1x+2x+3......x + (n-1)is prime.

Then I wrote this out and the answer was so clear:

( 6 x 5 x 4 x 3 x 2 x 1 ) + 2

( 6 x 5 x 4 x 3 x 2 x 1 ) + 2 + 1

( 6 x 5 x 4 x 3 x 2 x 1 ) + 2 + 2

( 6 x 5 x 4 x 3 x 2 x 1 ) + 2 + 3

( 6 x 5 x 4 x 3 x 2 x 1 ) + 2 + 4

( 6 x 5 x 4 x 3 x 2 x 1 ) + 2 - will always be divisible by 2

( 6 x 5 x 4 x 3 x 2 x 1 ) + 3 - will always be divisible by 3

( 6 x 5 x 4 x 3 x 2 x 1 ) + 4 - will always be divisible by 4

( 6 x 5 x 4 x 3 x 2 x 1 ) + 5 - will always be divisible by 5

( 6 x 5 x 4 x 3 x 2 x 1 ) + 6 - will always be divisible by 6

When I looked at the book it shows it like this which is even clearer. It factors out the number that this number is divisible by thus proving that this number is not prime.

2 ( 6 x 5 x 4 x 3 x 1 ) + 1

3 ( 6 x 5 x 4 x 2 x 1 ) + 1

The next theorem I will work on is one that no one has yet solved. But why not, I'll give it a shot. Show that there are an infinite number of twin primes.

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