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Math Help - My guess ... easy proof ?

  1. #1
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    My guess ... easy proof ?

    This is my guess , i hope that's true .

    Let's say a triangle is a Heron triangle if the lengths of its three sides as well as its area
    are integers.

    I guess : If  p ,q are primes such that  p^2 + 1 = 2q , then all the non-isosceles Heron triangles with p as the length of one of the sides are right-angled . Therefore , we can only find out one or two possible Heron triangle(s) for it .

    For example , let  p = 5 we have  5^2 + 1 = 2(13)

    Then the only Heron triangles are  (3,4,5) ~ ,~ (5,12,13) ?
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  2. #2
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    How about \left\{13, 14, 15\} or \left\{12, 35, 37\}?

    EDIT (Oops!):

    13^2+1 = 170 = 2(75) and 75 is not prime.
    37^2+1 = 1370 = 2(685) and [LaTeX ERROR: Convert failed] is not prime.
    Last edited by TheCoffeeMachine; July 22nd 2010 at 06:52 AM.
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  3. #3
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    Oh , i missed a thing so what i guessed is not necessarily correct !

    For example ,  29^2 + 1  = 2(421) but  (29,52,69) is not right-angled .
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  4. #4
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    Had it been true, it would have been a curiously nice result; and we would have called it Simplependulum's
    postulate (because it oddly reminded me of Bertrand's postulate). Keep guessing, my friend; keep guessing!
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