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Thread: Pythagorean Triples

  1. November 1st 2010, 05:18 PM #1
    MATNTRNG
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    Pythagorean Triples

    Find all of the Pythagorean triples that consist of consecutive positive integers. Prove that you have found them all.
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  2. November 1st 2010, 09:02 PM #2
    Soroban
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    Hello, MATNTRNG!

    Find all of the Pythagorean triples that consist of consecutive positive integers.
    Prove that you have found them all.

    Let the triple be: . (a,\:a+1,\:a+2)

    Then we have: . a^2 + (a+1)^2 \;=\;(a+2)^2

    . . . . . . . . . a^2 + a^2 + 2a + 1 \;=\;a^2+4a+4

    . . . . . . . . . . . . a^2 - 2a - 3 \:=\:0

    . . . . . . . . . . (a+1)(a-3) \:=\:0

    . . . . . . . . . . . . . . . . . . . a \:=\:\text{-}1,\:3


    Since \,a is a positive integer, a = 3.

    Therefore, the only Pythagorean triple is: . (3,\,4,\,5)
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