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    MHF Contributor alexmahone's Avatar
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    Sum of squares

    Prove that there are infinitely many positive integers n such that n(n + 1) can be expressed as a sum of two positive squares in at least two different ways. (Here a^2 + b^2 and b^2 + a^2 are considered as the same representation.)
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    Quote Originally Posted by alexmahone View Post
    Prove that there are infinitely many positive integers n such that n(n + 1) can be expressed as a sum of two positive squares in at least two different ways. (Here a^2 + b^2 and b^2 + a^2 are considered as the same representation.)
    Hint: If A = a^2+b^2 and B=c^2+d^2 then AB = (ac-bd)^2 + (ad+bc)^2.
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    Hello, Alex!

    Prove that there are infinitely many positive integers n such that n(n + 1)
    can be expressed as a sum of two positive squares in at least two different ways.
    (Here a^2 + b^2 and b^2 + a^2 are considered as the same representation.)

    ThePerfectHacker hinted at an interesting identity,
    . . one I had run across many years ago.

    (a^2+b^2)(c^2+d^2) \;=\;\begin{Bmatrix}(ac-bd)^2 + (ad + bc)^2 \\ (ac+bd)^2 + (ad-bc)^2 \end{Bmatrix}


    Let c^2 \:=\:a^2+b^2
    . . and there are an infinite number of Pythagorean triples.

    And we have: . c^2(c^2+d^2)

    Let d = 1 and we have: . c^2(c^2+1) which has the form: n(n+1)


    Example: .use 3^2+4^2\:=\:5^2\quad \begin{bmatrix} a\:=\:3 \\ b\:=\:4 \\ c\:=\:5 \\ d\:=\:1\end{bmatrix}

    (3^2+4^2)(5^2+1^2) \;=\;\begin{Bmatrix}(3\cdot5-4\cdot1) + (3\cdot1 + 4\cdot5)^2  \\(3\cdot5 + 4\cdot1) + (3\cdot1 - 4\cdot5)^2 \end{Bmatrix}

    . . . . . . (25)(26) \;=\;\begin{Bmatrix}11^2 + 23^2 \\ 19^2 + 17^2\end{Bmatrix} . . . all equal to 650



    Example: use 5^2+12^2 \:=\:13^2\quad\begin{bmatrix}a \:=\: 5 \\ b \:=\: 12 \\ c \:= \:13 \\ d \:=\: 1\end{bmatrix}

    (5^2+12^2)(13^2+1^2) \;=\;\begin{Bmatrix}(5\cdot13 - 12\cdot1)^2 + (5\cdot1 + 12\cdot13)^2 \\ (5\cdot13 + 12\cdot1)^2 + (5\cdot1 - 12\cdot13)^2 \end{Bmatrix}

    . . . . . . (169)(170) \;=\;\begin{Bmatrix}53^2 + 161^2 \\ 77^2 + 151^2\end{Bmatrix} . . . all equal to 28,730

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