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!

For any Pythagorean triple, the number of primes under a + b + c must

  1. Oct 29, 2013 #1
    be no more than c? In fact, only for the first triple does equality hold. Upon examining some of the triples, I noticed this must be true. However, I'm having a hard time explaining why. Is there a good explanation for this? Many thanks!
  2. jcsd
  3. Oct 29, 2013 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Since c is larger than a or b, you're basically saying the number of primes smaller than 3c is less than c... for c sufficiently large this is because the number of primes smaller than n is log(n). So the only worry would be that for c small you could have a counterexample and it just turns out there isn't one I guess. There might be a more solid reason but I would guess this is probably all that's happening.
  4. Oct 30, 2013 #3
    the number of primes smaller than n is approximately n/log(n), or more precisely:

    lim n→∞ (pi(n) log (n)) / n = 1

    where pi(n) is the number of primes smaller than n. (prime number theorem)

    You don't really need the prime number theorem here. If you only consider division by 2,3 and 5 it's easy to see that pi(n)< (8/30)n + 8 (because n mod 30 must be in {1,7,11,13,17,19,23,29})
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook