# Math Help - Infinitely many primes

1. ## Infinitely many primes

Is there a way to prove there is are infinitely many primes without doing a proof by contradiction?

2. Originally Posted by dwsmith
Is there a way to prove there is are infinitely many primes without doing a proof by contradiction?
A famous way is to show $\sum_{p \text{ prime}}\frac1p$ diverges.

3. Originally Posted by chiph588@
A famous way is to show $\sum_{p \text{ prime}}\frac1p$ diverges.
That harmonic series of course diverges since p=1 but I am confused on how showing divergence proves there are infinitely many primes.

4. Originally Posted by dwsmith
That harmonic series of course diverges since p=1 but I am confused on how showing divergence proves there are infinitely many primes.
This isn't the harmonic series. We're only summing over the prime numbers. i.e. $\sum_{p \text{ prime}}\frac1p = \frac12+\frac13+\frac15+\frac17+\frac{1}{11}+\frac {1}{13}+\cdots$

This proves infinitely many primes because if there were only finitely many primes, then $\sum_{p \text{ prime}}\frac1p$ couldn't possibly diverge.

5. Originally Posted by chiph588@
This isn't the harmonic series. We're only summing over the prime numbers. i.e. $\sum_{p \text{ prime}}\frac1p = \frac12+\frac13+\frac15+\frac17+\frac{1}{11}+\frac {1}{13}+\cdots$

This proves infinitely many primes because if there were only finitely many primes, then $\sum_{p \text{ prime}}\frac1p$ couldn't possibly diverge.
Is there a specific test I would use then show divergence and hence conclude there are infinitely many primes?

6. Originally Posted by dwsmith
Is there a specific test I would use then show divergence and hence conclude there are infinitely many primes?
Proof that the sum of the reciprocals of the primes diverges - Wikipedia, the free encyclopedia

I don't really like this proof because the first step takes the log of infinity, but it will give you the right idea and can be tweaked a bit to be valid.

7. Originally Posted by chiph588@
Proof that the sum of the reciprocals of the primes diverges - Wikipedia, the free encyclopedia

I don't really like this proof because the first step takes the log of infinity, but it will give you the right idea and can be tweaked a bit to be valid.
I don't really like any of those.

8. Originally Posted by dwsmith
I don't really like any of those.
Maybe you'll like this one better: Goldbach's Proof of the Infinitude of Primes (1730)

9. Originally Posted by dwsmith
Is there a way to prove there is are infinitely many primes without doing a proof by contradiction?
Euclid's proof can be modified into a constructive proof that there is no largest prime.

CB

10. Here's a proof I thought about a while back (which probably exists elsewhere!). It's not rigorous, as one doesn't really make precise the idea of putting "half" of the integers in a set. However, it's still an interesting reasoning.

Suppose there are only $n$ primes, the greatest being $p$.
Half of the integers are not divisible by 2.
Two-thirds of those are not divisible by $3$.
Four fifths of those are not divisible by $5$.
All the way to : $(p-1)/p$ths are not divisible by $p$.

Hence the "proportion" of integers not divisible by any one of $2, 3, \dots, p$ is equal to $\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\dots \left(1-\frac{1}{p}\right)$, which is greater than $0$, which is absurd because every integer must be a multiple of one of $2, 3, \dots, p$, which we've supposed to be the only primes.

The reasoning also shows informally that $\prod_{p}\left(1-\frac{1}{p}\right)=0$; expanding the product yields a nice Dirichlet series for the Möbius $\mu$ function. Note also that the product is $1/\zeta(1)$.

11. There is a famous relationship due to Euler: $1+\frac12+\frac13+\dots$(continues forever) $=(1+\frac12+\frac1{2^2}+\dots)\times(1+\frac13+\fr ac1{3^2}+\dots)\times\dots\times(1+\frac1p+\frac1{ p^2}+\dots)\times\dots$

The left-hand side is the harmonic series that diverges. The right-hand side is a product in which each member is of finte value (for example: $1+1/2+1/2^2+\dots=2, 1+1/3+1/3^3+\dots=3/2$. In general, $1+1/p+1/p^2+\dots=p/(p-1)$).

The right-hand side, then must be an infinte product because the harmonic series is divergent.

12. Originally Posted by chiph588@
Maybe you'll like this one better: Goldbach's Proof of the Infinitude of Primes (1730)
This was beautiful ! Thanks

I have a much less consequential question (related to the approach used in the Goldbach's proof)

Let a,b be co-prime

Let
a1 = a
a2 = a1 + b
a3 = a1a2 + b
a4 = a1a2a3 + b
....
...
.

Prove that (ai,aj) are also co-prime for any i and j

Now I struggled with this one - finally could do it using induction. But I'm looking for a better way to do it? Can anyone help plz? I will be more than happy to submit my proof if anyone needs to see that.