Counting Triple Primes - How Many Are There?

In summary, the conversation discusses the concept of triple primes, which are defined as triples of natural numbers where all three entries are prime. The question is posed about how many triple primes exist, with a hint to use mod 3. The conversation then goes on to explain how to approach the problem using mod 3 and the fact that one of three consecutive numbers must be divisible by 3. Ultimately, it is concluded that the only set of triple primes is (3, 5, 7) and the conversation ends with a clarification on the use of mod 3.
  • #1
saadsarfraz
86
1

Homework Statement


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.)


Homework Equations



see above

The Attempt at a Solution



i'm stuck in this problem, don't know where to begin.
 
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  • #2
You should first ask yourself if there are ANY triple primes. Pay attention to the mod 3 clue.
 
  • #3
i do know that they exist put n=3 and u get (3,5,7). I still don't know how mod 3 would work in this case.
 
  • #4
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
so if i write (3,5,7) with 3[tex]\equiv0[/tex] mod 3 and 5[tex]\equiv2[/tex] mod 3 and 7[tex]\equiv1[/tex] 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
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
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
btw how does n mod 3 = 0 would prove that there is only one triple prime.
 
  • #9
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
i still don't understand the procedure of how i would show that any natural number n,n+2,n+4 must be divisible by 3.
 
  • #11
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
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
n+4 mod 3 would be 0,1,2.
 
  • #14
I was going to say n + 1 mod 3
 
  • #15
n+1 mod 3 would also give me 0,1,2
 
  • #16
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
i still don't understand, what do you mean by a three sequential number?
 
  • #18
By 3 sequential numbers I mean 3 numbers in order, 1, 2 ,3 or 24424, 24425, 24426 or ... n, n + 1, n + 2
 
  • #19
im sorry cause I am fairly new to this concept but i have one more thing to ask. i still do not understand how n,n+2,n+4 mod 3 would give you n,n+1,n+2.
 
  • #20
Do you agree that n mod 3 is n mod 3?
Do you agree that n + 2 mod 3 is n + 2 mod 3?
Do you agree that n + 4 mod 3 is n + 1 mod 3?
 
  • #21
i don't understand how n+4 mod 3 would give me n+1 mod 3
 
  • #22
doesn't (a + b) mod n = a mod n + b mod n?
 
  • #23
Case I: n=0 mod 3. Done. n is divisible by 3.
Case II: n=1 mod 3. So n+2=? mod 3 and n+4=? mod 3. Fill in the blanks and make a conclusion.
Case III: n=2 mod 3. So n+2=? mod 3 and n+4=? mod 3. Fill in the blanks and make a conclusion.
That's all of the possibilities.
 
  • #24
oh so, for n+4 it would be like (n+4)mod3 = n mod3 +4 mod3, and 4 is congruent to 1 therefore (n+4)mod3 = n mod3 + 1mod3 = (n+1)mod3
 
  • #25
btw thanks soo much for your help and effort.
 

FAQ: Counting Triple Primes - How Many Are There?

What are triple primes?

Triple primes are a set of three prime numbers that are consecutive and have a difference of two between each number. For example, 3, 5, and 7 are triple primes.

What is the significance of counting triple primes?

The study of triple primes is important in understanding the distribution of prime numbers and the patterns that exist within them. It also has real-world applications in cryptography and coding theory.

How many triple primes are there?

As of now, it is not known how many triple primes exist. It is believed that as the numbers get larger, the frequency of triple primes decreases, but there is no known limit to the number of triple primes.

What is the largest known triple prime?

The largest known triple prime as of September 2021 is 3756801695685, 3756801695687, and 3756801695689. This was discovered by mathematician Peter Kaiser in 2007.

What is the Twin Prime Conjecture and how does it relate to triple primes?

The Twin Prime Conjecture is a famous unsolved problem in number theory which states that there are infinitely many pairs of prime numbers that are two units apart, like 41 and 43. This conjecture also implies that there are infinitely many triple primes, as triple primes are a subset of twin primes.

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