1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Number of theory

  1. Oct 13, 2012 #1
    1. The problem statement, all variables and given/known data

    if p and p+2 are twins prime and p+1 is triangular number, then find all twin primes.

    2. Relevant equations

    Tn is a triangular number if Tn=1+2+...+n = n(n+1) / 2

    3. The attempt at a solution
    p+1 = n(n+1) / 2 because p+1 is a triangular number
    p , p+1 , p+2 are terms in succession
    p is prime, p+2 is prime so 3 divide p+1 --> 3/p+1
    sure the p+1 is even number because is between of two prime
    so p+1 has the form 6k
    i don't how to continue.. can someone help me??
    Last edited: Oct 13, 2012
  2. jcsd
  3. Oct 15, 2012 #2
    I think the statement of the problem should be:

    Find all twin primes p and p + 2 such that p + 1 is a triangular number.

    You showed that if p and p + 2 are primes, then p + 1 is divisible by 6. That statement is true, but I couldn't see how it helps to solve the problem. Here's how I came up with a solution. Let T(n) = n(n+1)/2 be the nth triangular number. Create a table as follows:

    n& T(n)& T(n) - 1& T(n) + 1\\
    1& 1& 0& 2\\
    2& 3& 2& 4\\
    3& 6& 5& 7\\
    4& 10& 9& 11\\

    and so on. Note that for n = 3, we obtain the twin primes 5 and 7 that satisfy the required condition. Now extend the table for several more values of n (up to n = 10 for example). Do you find any more twin primes? Do you see a pattern (especially in the values of T(n) - 1)? If so, state the pattern precisely and prove it.

    Please post again if you have any questions or would like additional hints.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook