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Thread: 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 $\displaystyle p ,q $ are primes such that $\displaystyle p^2 + 1 = 2q $ , then all the non-isosceles Heron triangles with $\displaystyle p$ 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 $\displaystyle p = 5 $ we have $\displaystyle 5^2 + 1 = 2(13) $

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

    EDIT (Oops!):

    $\displaystyle 13^2+1 = 170 = 2(75)$ and $\displaystyle 75$ is not prime.
    $\displaystyle 37^2+1 = 1370 = 2(685)$ and $\displaystyle 685$ is not prime.
    Last edited by TheCoffeeMachine; Jul 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 , $\displaystyle 29^2 + 1 = 2(421)$ but $\displaystyle (29,52,69)$ 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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