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 July 4th, 2009, 07:46 AM #1 Senior Member   Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0 Primes inside the arithmetic progression Prove that among progression 3, 7, 11, 15, 19, ... there is infinitely many primes.
 July 4th, 2009, 09:23 AM #2 Senior Member   Joined: Jun 2009 Posts: 150 Thanks: 0 Re: Primes inside the arithmetic progression Call such a number "3 mod 4" ... the rest of the odd numbers are "1 mod 4" ... Note that the product of two "1 mod 4" numbers is again a "1 mod 4" number. If there were no "3 mod 4" primes, then there could be no "3 mod 4" numbers at all. A contradiction. Therefore, there is at least one "3 mod 4" prime. (By itself, this is not very useful, because we can easily write down such a prime.) Now: soup up that argument: assume there are only finitely many "3 mod 4" primes, then get a contradiction from that.
 July 4th, 2009, 09:29 AM #3 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Primes inside the arithmetic progression It's not hard to modify Euclid's theorem to work in this case. Suppose that the only primes of this form are 3, 7, ..., k. Take their product and add 4. Can this be divisible by any of the numbers on your list? Can it be the product of primes = 1 mod 4?
July 5th, 2009, 05:40 AM   #4
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Re: Primes inside the arithmetic progression

Quote:
 Suppose that the only primes of this form are 3, 7, ..., k. Take their product ...
That's exactly the way I started with.
Quote:
Actually, I will need to add 4 or 2 depending on the product. Won't I?
Quote:
 Can this be divisible by any of the numbers on your list?
No way.
Quote:
 Can it be the product of primes = 1 mod 4?
Here I'm stuck.

July 5th, 2009, 08:59 AM   #5
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Re: Primes inside the arithmetic progression

Quote:
 Originally Posted by lime Actually, I will need to add 4 or 2 depending on the product. Won't I?
Good catch.

Quote:
Originally Posted by lime
Quote:
 Can it be the product of primes = 1 mod 4?
Here I'm stuck.
Does 1 * 1 = 3 (mod 4)?

 July 5th, 2009, 09:52 AM #6 Senior Member   Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0 Re: Primes inside the arithmetic progression Can't believe I was so silly. Thanks!

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