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Re: [PrimeNumbers] Mersenne Geometry

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  • Jud McCranie
    ... It s a little late at night for me to be thinking about this, but I think the answer is no. Squares are 0 or 1 mod 3. If the first leg is 2^n-1, the
    Message 1 of 2 , Dec 26, 2001
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      At 11:46 PM 12/26/2001 -0500, SWagler@... wrote:
      >If a right triangle is constructed with one leg equal to a Mersenne prime and
      >the other leg equal to the following power of 2, then the area of the
      >triangle is equal to a perfect number. The first such triangle is the special
      >3-4-5 triangle which is also a Pythagorean triangle (all sides integer). None
      >of the rest of the triangles that I could check were Pythagorean. Is it
      >possible that any other triangles could be Pythagorean?

      It's a little late at night for me to be thinking about this, but I think
      the answer is no. Squares are 0 or 1 mod 3. If the first leg is 2^n-1,
      the square of the hypotenuse is 2^(2n+1)-2^(n+1)+1. If n is odd, this is
      congruent to 2 mod 3. So if n is odd, the square of the hypotenuse can't
      be a perfect square. All Mersenne primes after the first one have n odd,
      so no more result in Pythagoran triangles.


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      | Jud McCranie |
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