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Math Help - ordered integer pair,s (x,y)

  1. #1
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    ordered integer pair,s (x,y)

    (1) If (1!)^2+(2!)^2+(3!)^2+.......................+(x!)^  2 = y^2. Then ordered integer pair,s of (x,y) , where (x,y)\in \mathbb{N}

    (2) If (1!)^3+(2!)^3+(3!)^3+.......................+(x!)^  3 = y^3. Then ordered integer pair,s of (x,y) , where (x,y)\in \mathbb{N}
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  2. #2
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    Re: ordered integer pair,s (x,y)

    (1,1) is one such ordered pair for both of those. I ran a check in Mathematica, and up to x=1000, there are no other valid pairs for either of them. You might try continued fractions...

    \cfrac{\cfrac{\cfrac{y^2-1}{2^2}-1}{\cdots}-1}{x^2} = 1 (for part (1))

    \cfrac{\cfrac{\cfrac{y^3-1}{2^3}-1}{\cdots}-1}{x^3} = 1 (for part (2))
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