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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.

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    $\begingroup$ Hint : All primes but $2$ are odd. $\endgroup$ – Vincent Dec 19 '16 at 12:21
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    $\begingroup$ 1) prove that all primes > 2 are odd ---- 2) prove that the gap between any two odd numbers is even (hint all odd numbers are of the form 2a + 1 where a is a whole number) $\endgroup$ – Cato Dec 19 '16 at 12:42
  • $\begingroup$ $2n+1 - (2k+1) = 2(n-k)$ $\endgroup$ – Widawensen Dec 19 '16 at 14:33
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    $\begingroup$ I don't think this question should be reopened yet. $\endgroup$ – Brian Tung Dec 19 '16 at 18:23
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$1$. 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. :) :)

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    $\begingroup$ 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 $\endgroup$ – Cato Dec 19 '16 at 12:47
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    $\begingroup$ @Cato, it is done. $\endgroup$ – Vidyanshu Mishra Dec 19 '16 at 12:57
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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.

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    $\begingroup$ 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." $\endgroup$ – TMM Dec 19 '16 at 12:31
  • $\begingroup$ 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. $\endgroup$ – I_Really_Want_To_Heal_Myself Dec 19 '16 at 12:34
  • $\begingroup$ See also math.stackexchange.com/questions/240/… and mathoverflow.net/questions/12342/…. 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. $\endgroup$ – TMM Dec 19 '16 at 12:47
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All primes other than $2$ are odd, hence gap between them is even (as difference of two odd numbers is always even).

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