All other prime gaps (except between $2$ and $3$) are even.

Question

How can one show that except for the first prime gap (between $2$ and $3$), all prime gaps are even?

Prime gap means the difference of two consecutive primes. For example: $5-3=2$ (even) or $17-13=4$ (even) or $23-19=4$ (even).

I have no idea how to start.

Hint : All primes but $2$ are odd. – Vincent Dec 19 '16 at 12:21 — Vincent, Dec 19 '16 at 12:21
I don't think this question should be reopened yet. – Brian Tung Dec 19 '16 at 18:23 — Brian Tung, Dec 19 '16 at 18:23

Vidyanshu Mishra · Accepted Answer · 2016-12-22T16:26:33.977

$1$. The only even prime is $2$ and all other are odd primes.

$2$. Difference between two odd numbers is always even.

First posted as hint but now @Cato asked for a proof so here We go:

Proof of $1$:

Suppose that there is another prime which is even (other than $2$), then it must be of the form $2k$ where $k$ being a positive integer, so that prime number can be written as $2 \times k$ which is enough to say that it is composite (Or not prime). A contradiction.

Proof of $2$.

As known (I don't think it also need a proof), every odd number is of the form $2k+1$, where $k$ is an integer. We just have to calculate the difference b/w two odd primes (Remember the word odd) so let first one be $2k_1+1$and second one be $2k_2+1$. Difference $= (2k_2+1-(2k_1+1))= 2k_2-2k_1=2(k_2-k_1)=$ even number.

I think it is enough. :) :)

ideally he needs to prove all the facts he uses in the proof - so 2 is the only prime number (needs a proof) - odd - odd = even - possibly needs some proof — Cato, Dec 19 '16 at 12:47

score 5 · Answer 2 · answered Dec 19 '16 at 12:22

5

Since all prime numbers except $2$ are odd, then if a prime gap $g$ between primes $p_1,p_2$ would be odd, then $p_2=p_1+g$ would be the sum of two odd numbers which is even, and since the only prime which is even equals $2$, thus we get the desired contradiction.

answered Dec 19 '16 at 12:22

I_Really_Want_To_Heal_Myself

443

2

Why the proof by contradiction when the "constructive" proof is just as simple? It's a bit like saying "Suppose P does not hold. But we know that P does hold because of this other property, hence a contradiction. So P is true." – TMM Dec 19 '16 at 12:31
You're right, but proofs by contradiction are easier to understand for me at least. Guess it's just the way I think, but you're right of course, constructive proof is really simple. – I_Really_Want_To_Heal_Myself Dec 19 '16 at 12:34
See also http://math.stackexchange.com/questions/240/are-the-proofs-by-contradiction-weaker-than-other-proofs and http://mathoverflow.net/questions/12342/reductio-ad-absurdum-or-the-contrapositive. If there's a choice between a direct proof and one by contradiction, the general consensus seems to be that a direct proof should be preferred for various reasons. – TMM Dec 19 '16 at 12:47

score 5 · Answer 3 · edited Dec 20 '16 at 09:36

5

All primes other than $2$ are odd, hence gap between them is even (as difference of two odd numbers is always even).

edited Dec 20 '16 at 09:36

Vidyanshu Mishra

10,143

answered Dec 19 '16 at 14:29

Blaise Thunderstorm

1,368

All other prime gaps (except between $2$ and $3$) are even.

3 Answers3