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Math Help - Quickie #12

  1. #1
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    Quickie #12

    For what n is \sum^n_{k=1} k! a square?

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  2. #2
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    Quote Originally Posted by Soroban View Post
    For what n is \sum^n_{k=1} k! a square?

    Certainly for 1, 3, although for all n in general I'd start by using the fact

    n^2 = summation(2k-1, k = 1...n)

    and go from there.
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  3. #3
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    I have a feeling 20 is one...
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  4. #4
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    Quote Originally Posted by anthmoo View Post
    I have a feeling 20 is one...
    No, 20 is not one. I checked with Maple.

    You end up with 33sqrt(2.3519995354562*10^15)
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  5. #5
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    Hello, AfterShock!

    You found the only two solutions . . .


    Note that squares must end in: 0,\,1,\,4,\,5,\,6,\,9


    List the first few sums:

    \begin{array}{cccc} 1! & = & \boxed{1}\\ 1!+2! & = & 3\\ 1!+ 2!+3! & = & \boxed{9}\\ 1!+2!+3!+4! & = & 33\end{array}

    Since 5!,\,6!,\,7!,\,\hdots all end in 0,
    . . all subsequent sums will end in 3
    . . and hence cannot be squares.

    Therefore: n = 1,\,3 are the only solutions.

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