Primary Problem 2 (Posted on 2012-12-09)

	Submitted by Math Man
Rating: 5.0000 (1 votes)
Solution:	(Hide)
Suppose p1, p2, ..., pn are all of the primes of the form 4x+1. Let P=(2p1p2...pn)2+1. Since P is one more than a square, its only possible prime factors are 2 and the primes of the form 4x+1. P is odd, so it is not divisible by 2. Therefore, P can only have prime factors of the form 4x+1. However, it cannot be divisible by any of p1, p2, ..., pn. Therefore, there are infinitely many primes of the form 4x+1. Suppose p1, p2, ..., pn are all of the primes of the form 4x-1. Let N=4p1p2...pn-1. P is of the form 4x-1, so it is not divisible by 2. If all of P's factors were of the form 4x+1, then P would be of the form 4x+1. Therefore, P must have at least one prime factor of the form 4x-1. However, it cannot be divisible by any of p1, p2, ..., pn. Therefore, there are infinitely many primes of the form 4x-1.

	Subject	Author	Date
	possible approach	broll	2012-12-10 11:43:31
	re(2): Too simplistic?	Math Man	2012-12-10 11:23:20
	re: Too simplistic?	Charlie	2012-12-10 10:15:40
	Too simplistic?	brianjn	2012-12-10 03:51:49