Monday, April 23, 2007

Why does x = (n+1)! =2 produce n consecutive non-prime numbers?

I am going to try working with the example of n=5 to see if I can figure out why this is true in the general case:

Suppose n is a positive integer. Let x = (n + 1)! +2
Show that none of the numbers x, x+1, x+2, x+3, ......x + (n-1) is prime.

So, what about this for the case of n=5 insures that non-prime numbers result?
( 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


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

This number ( 3 x 2 x 5 x 2 x 2 x 3 x 2 x 1 ) which is (n+1)! in the case of n = 5 is where I will turn my attention today while thinking about how to prove this theorem.

cheers,
alex

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