Infinite prime number proof?

Question

What type of proof would be acceptable if I were to prove that the set {3,7,11,15,19,.........} contained infinitely many primes? Is there really an airtight solution to this?

Answer 1

Start with a few known primes...

2, 3, 5, 7, 11. Take these and multiply them and add 1. That will also be a prime because it will have none of the earlier primes as a factor.

Because you can repeat this process ad infinitem, that proves that there are an infinite number of primes.

Alternatively you could prove it by contradiction as follows:

Assume you have found a finite set of primes P. Now multiply all the elements of that set and add 1. This new number is prime because the remainder when dividing by any of the prior primes will always be 1 (none are 0). And that number is not included in the set, thereby contradicting your assumption that there are a finite number of primes.

Answer 2

Attention: read the question more carefully! The asker doesn't want a proof that there are infinitely many primes. The asker wants a proof that there are infinitely many primes in the set {3, 7, 11, 15, 19....}.

Alternatively: are there infinitely many primes congruent to 3 mod 4?

It's easy to guess that there are infinitely many such primes, but I don't know a proof of it.

Answer 3

The first person to prove this in the general case was Dirichlet, in 1837, so even the great Carl Friedrich Gauss must never have constructed a proof of it that satisfied him. I rather think that unless you have a Ph.D. in number theory, or you are an undiscovered genius, your proof will be somehow flawed. Nevertheless, it may be that the special case 4n+3 has some feature which allows a simpler proof than Dirichlet's general case an+b.

P.S. obviously Dirichlet's proof was for an+b where a and b have no common factor.

Answer 4

Suppose there are only finitely many prime numbers and let Pr = {p_1,p_2.....p_n} be the set of such primes.Put p = p_1 * p_2 *....p_n. Then, each p_i, i=1, 2,...n divides p. According to the Fundamental Theorem of Arithmetic, p+1 is a product of prime numbers, which implies there's a p_i in Pr that divides p+1. Since this p_i also divdes p, it follows p_i divides (p+1) - p =1. But since p_i >=2, this is a contradiction, which proves there are infinitely many prime numbers and Pr is an infinite set.

Answer 5

2, 3, 5, 7, 11 Then multiply by ONE. No, theres not really an "airtight" solution to this. Just logical intelligence to see the answer... very simple.

Answer 6

If you can prove that there are an infinite number of twin primes, you have also proved this theorem too

Answer 7

yeh i think that add 4

Answer 8

probably just add 4