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Math Help - Pythagorean triplet

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  2. #2
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    Hello, MATNTRNG!

    I have a very primitive solution . . .


    Find the only Pythagorean triplet, \{a, b, c\}, for which a + b + c \:=\: 1000
    All Pythagorean triples are generated by: . \begin{Bmatrix}a &=& |m^2-n^2| \\ b &=& 2mn \\ c &=& m^2+n^2 \end{Bmatrix}\quad\text{for integers }\:m, n\,>\,0


    We have: . a+b+c \;=\;(m^2-n^2) + (2mn) + (m^2+n^2) \;=\;1000

    . . which simplifies to: . m(m+n) \:=\:500


    There are only seven solutions.

    . . \begin{array}{|c|c || c|c|c|}<br />
m & n & a & b & c \\ \hline<br />
1 & 499 & 249,\!000 & 998 & 249,\!002 \\<br />
2 & 248 & 61,\!500 & 992 & 61,\!508 \\<br />
4 & 121 & 14,\!625 & 968 & 14,\!657 \\<br />
5 & 95 & 9000 & 950 & 9050 \\<br />
8 & 117 & 13,\!625 & 1872 & 13,\!753 \\<br />
10 & 40 & 1500 & 800 & 1700 \\<br />
20 & 5 & 375 & 200 & 425 \\ \hline \end{array}


    And only the last triplet \{375,\:200,\:425\} has a sum of 1000.

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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by MATNTRNG View Post
    Find the only Pythagorean triplet, {a, b, c}, for which a + b + c = 1000.
     \begin{cases} a=m^2-n^2 \\ b=2mn \\ c=m^2+n^2 \end{cases}

    So  a+b+c = m^2-n^2+2mn+m^2+n^2 = 2m^2+2mn = 1000

    Therefore we want to solve  m^2+mn=500

     4m^2+4nm-2000 = 0

    Complete the square:  (2m+n)^2-2000-n^2 = 0

    Let  x=2m+n to get  n^2-x^2 = -2000

    Factor:  (n+x)(n-x) = -2^4\cdot5^3

    Now we just have finitely many cases to solve. For example solve  \begin{cases} n+x=4 \\ n-x=-500 \end{cases}

    I'll let you solve all of these out, but you should only get one solution with  m>n>0 , namely  (m,n) = (20,5)

    Now plug these into the formulas for  a,b,c to get  \begin{cases} a=375 \\ b=200 \\ c=425 \end{cases}
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    The proof of a=m^2-n^2,b=2mn,c=m^2+n^2 assumed that (a,b)=1. Although it's easily seen that the formula still applies if (a,b) is a perfect square, but there's a potential that you'll miss some solution (although here it doesn't happen) if you merely consider a=m^2-n^2,b=2mn
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