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Math Help - Pythagorean triple in (a,a+1,c)

  1. #1
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    Pythagorean triple in (a,a+1,c)

    Pythagorean triple is (a,b,c) satisfies a^2+b^2=c^2

    Find all Pythagorean triple in (a,a+1,c)

    the first two are (3,4,5) and (20,21,29)

    thanks!!!
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  2. #2
    Super Member PaulRS's Avatar
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    We require: a^2+(a+1)^2=c^2

    Expand: 2a^2+2a+1=c^2

    Now multiply by 2: 4a^2+4a+2=2c^2

    So: 4a^2+4a+1+1=2c^2 and note that: 4a^2+4a+1=(2a+1)^2

    We have: <br />
\left( {2a + 1} \right)^2  + 1= 2c^2 <br />

    Set x=2a+1 and y=c

    So our condition turns into: x^2+1=2y^2

    And this is a Pell Type equation ! (Read here, if you have any doubts, ask )
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  3. #3
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    Hello, chipai!

    Pythagorean triple is (a,b,c) satisfies a^2+b^2\:=\:c^2

    Find all Pythagorean triple of the form: (a,a+1,c)

    The first two are (3,4,5) and (20,21,29).
    The first few are:

    . . \begin{array}{ccc} a & b & c \\ \hline<br />
3 & 4 & 5 \\ 20 & 21& 29 \\ 119& 120& 169 \\ 696 & 697 & 985 \\ 4059 & 4060 & 5741 \end{array}


    The recursive formulas are:

    . . \begin{array}{ccc}a_n &=& 6a_{n-1} - a_{n-2} + 2 \\ \\[-3mm]<br />
c_n &=& 6c_{n-1} - c_{n-2}\qquad \end{array}

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