# Triple primes

1. Jan 19, 2009

1. The problem statement, all variables and given/known data
Here's the problem. We define the triple primes as triples of natural numbers (n,n+2,n+4) for which all three entries are prime. How many triple primes are there? (Hint:mod 3.) (By way of contrast, it is not yet known whether the twin primes-that is, pairs (n,n+2) with both entries prime-form an infinite collection.)

2. Relevant equations

see above

3. The attempt at a solution

i'm stuck in this problem, don't know where to begin.

2. Jan 19, 2009

### Dick

You should first ask yourself if there are ANY triple primes. Pay attention to the mod 3 clue.

3. Jan 20, 2009

i do know that they exist put n=3 and u get (3,5,7). I still dont know how mod 3 would work in this case.

4. Jan 20, 2009

### Dick

Oh, yeah. That one. Notice that one of those primes is divisible by three. Write down any 3 consecutive odd numbers and notice that one of them is divisible by 3. Can you give the reason why using a 'mod 3' argument? Then ask yourself how many primes are divisible by 3?

5. Jan 20, 2009

so if i write (3,5,7) with 3$$\equiv0$$ mod 3 and 5$$\equiv2$$ mod 3 and 7$$\equiv1$$ mod 3 how does that help? and there are no primes which are divisible by 3 unless it would not be a prime number.

6. Jan 20, 2009

### Dick

Just do the general case. Take n,n+2,n+4. n mod 3 is either 0 (in which case you are already done),1 or 2. Consider the other two cases and figure out what n+2 and n+4 are.

7. Jan 20, 2009

n+2 is going to be mod 5 with 0,1,2,3,4. and n+4 is going to be mod 7 with 0,1,2,3,4,5,6. right?

8. Jan 20, 2009

btw how does n mod 3 = 0 would prove that there is only one triple prime.

9. Jan 20, 2009

### jgens

I think what Dick is trying to get you to do is to show that given any natural number n: n, n +2, or n + 4 one of them must be divisible by 3; hence, (3, 5, 7) must be the only set of triple primes.

10. Jan 20, 2009

i still dont understand the procedure of how i would show that any natural number n,n+2,n+4 must be divisible by 3.

11. Jan 20, 2009

so for any n, either n, n+2 or n+4 must be divisible by 3 which means either of them should be mod 3?

12. Jan 20, 2009

### NoMoreExams

Yes, you know that if you have k sequential numbers, one of them will be divisible by k. What's n + 4 mod 3?

13. Jan 20, 2009

n+4 mod 3 would be 0,1,2.

14. Jan 20, 2009

### NoMoreExams

I was going to say n + 1 mod 3

15. Jan 20, 2009

n+1 mod 3 would also give me 0,1,2

16. Jan 20, 2009

### NoMoreExams

Obviously but the thing I was hoping that you would notice is that you have n, n + 2, n + 4. If we do them mod 3 we get n, n + 1, n + 2 which is THREE sequential numbers so one of them MUST be divisible by 3.

17. Jan 20, 2009

i still dont understand, what do you mean by a three sequential number?

18. Jan 20, 2009

### NoMoreExams

By 3 sequential numbers I mean 3 numbers in order, 1, 2 ,3 or 24424, 24425, 24426 or ... n, n + 1, n + 2

19. Jan 20, 2009