Pairs of primes separated by a single number are called prime pairs

[EvLer]

here's one more:
pairs of primes separated by a single number are called prime pairs. Example: 17 and 19 are a pair. Prove that the number between prime pair is always divisible by 6 (assuming both numbers are greater than 6).

 

[AKG]

Suppose x, x+2 are prime pairs. We need to show x+1 is divisible by 6. Well it is divisible by 2, because if it weren't, both x and x+2 would be, and so they wouldn't be primes, but they are. It is also divisible by 3 because if it weren't, one of x and x+2 would be (since in any set of 3 consecutive numbers, one of them is divisible by 3), and so one of them wouldn't be prime, but they are. x+1 is divisible by 3 and 2, hence by 6.

 

[EvLer]

yup... that's how i solved it...

 

[Edgardo]

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 let's 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.
QED.
Please tell me whether my solution is understandable.

 

[croxbearer]

yes it is..

 

[ssj5harsh]

I solved it this way.
The solution should have something to do with number 6.
Nos. can only be of the form:
6k, 6k+1, 6k+2, 6k+3, 6k+4, 6k+5 (k is an integer)
Out of these the only types which can be primes are 6k+1 and 6k+5 (ie 6(k+1)-1).
Evidently two twin primes will be of the form, 6n-1 and 6n+1.(n is an integer)
The number between them is 6n, definitely divisible by 6.
Hence Proved

 

[Edgardo]

ssj5harsh, I don't understand your solution

 

[LarrrSDonald]

6k, 6k+1, 6k+2, 6k+3, 6k+4, 6k+5 (k is an integer)

He is essentially saying that 6k will be divisible by 2 and 3 (6 is, after all, a factor). Hence, 6k+2 will also be divisible by 2 (adding two doesn't change divisibility by 2), 6k+3 by 3, 6k+4 by 2 (adding +2+2). Thus, only 6k+1 and 6k+5 have the option of being prime. Thus, the number between them must be 6k, divisible by 6. I think.

 

[Edgardo]

Thanks for the explanation LarrrSDonald.

 

[MizardX]

here's one more:
pairs of primes separated by a single number are called prime pairs. Example: 17 and 19 are a pair. Prove that the number between prime pair is always divisible by 6 (assuming both numbers are greater than 6).

It's true for all prime-pairs except 3-5
Between 3 and 5 there's 4, which is, in fact, not divisible by 6

 

[Edgardo]

It's true for all prime-pairs except 3-5
Between 3 and 5 there's 4, which is, in fact, not divisible by 6

But EvLer said:

assuming both numbers are greater than 6

:tongue:

 

[ssj5harsh]

Hmm..

I realized something,
the sum of the two twin primes is always divisible by 12.
Not unexpected, but fascinating nevertheless.

 

[Curious3141]

I realized something,
the sum of the two twin primes is always divisible by 12.
Not unexpected, but fascinating nevertheless.

That would follow immediately from the fact that the middle number is a multiple of 6 : (6k-1) + (6k+1) = 12k.