Thread: Twin Prime

Prove that number between twin prime is divisible by 6.

To show that the number x between a prime pair is divisible by
6, I show that it is divisible by 2 and 3.
picture: p x (p+2)
The picture shows the prime pair p and (p+2) and the number x in between.

(i) Show that x is divisible by 2:
Either the number x is odd or even. We know that one number left from x is a
prime number (see picture) and this prime number can't be divided by 2, that
means it is odd. After this (odd) prime number follows an even number,
and this number is x.

(ii) Show that x is divisible by 3:
When we examine how the multiples of 3 are distributed, we discover the
following pattern:
3 4 5 6 7 8 9 10 11 12
3 - - 6 - - 9 - - 12 - - and so on. The two minus signs stand for
two numbers between the multiples of 3.
In general:
x - - (x+3) - - and so on.
There are TWO numbers between two multiples of 3, which can't be divided by 3 (the TWO minus signs).

Now lets examine the first picture:
p x (p+2)
with p and (p+2) the prime pair and x the number in between.

Since p is prime, it is not a multiple of 3 and therefore can be replaced by a minus sign:
p x (p+2) turns into - x (p+2).
Now the question is, whether the second minus sign is located LEFT or RIGHT from the minus sign.
Let's assume that the second minus sign is located right (replace x by a minus sign), therefore:
- x (p+2) turns into - - (p+2). But this can't be since (p+2) is not a multiple of 3.
Therefore the second minus sign must be located left:
- x (p+2) turns into - - x (p+2), which means that x is a multiple of 3.