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Square pyramidal numbers and Tetrahedral numbers

  1. Oct 29, 2009 #1
    There are square pyramidal numbers and tetrahedral numbers, defined

    Square pyramidal numbers = n ( n + 1 )( 2 n + 1) / 6
    1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, ...

    Tetrahedral numbers = n ( n + 1 )( n + 2 ) / 6

    1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, ...

    and I was wondering if there's a number(s) besides 1 that is both.
     
  2. jcsd
  3. Oct 29, 2009 #2
    I have to guess no. Purely for probabilistic reasons. For any number n, the probability that there is a pyramidal number equal to n(n+1)(n+2)/6 is ~1/n^2, falling off too fast as n -> infinity. So, for example, the probability of a hit for n>100 is ~0.01. Once we've checked the first 100 n's, we can be fairly sure that there won't be any hits beyond that.

    But I have no idea how to give a proper proof.
     
  4. Oct 29, 2009 #3
    The answer is no. You are looking for solutions of the equation:

    [tex]
    \frac {n(n+1)(2n+1)}{6} = \frac {n(n+1)(n+2)}{6}
    [/tex]

    Cancelling terms we get

    [tex]
    2n+1 = n+2
    [/tex]

    Solving for n you find n = 1 as the only solution
     
  5. Oct 29, 2009 #4

    CRGreathouse

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    The OP is looking for solutions to

    [tex]
    \frac {n(n+1)(2n+1)}{6} = \frac {m(m+1)(m+2)}{6}
    [/tex]
    not

    [tex]
    \frac {n(n+1)(2n+1)}{6} = \frac {n(n+1)(n+2)}{6}
    [/tex]
    .
     
  6. Oct 30, 2009 #5
    According to Mathworld, 1 is the only solution, and this fact was only proven in 1988 (so, no easy proof is forthcoming).
     
  7. Oct 30, 2009 #6
    really? Can you give a cite?
     
  8. Oct 30, 2009 #7

    CRGreathouse

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    MathWorld gives
    Beukers, F. "On Oranges and Integral Points on Certain Plane Cubic Curves." Nieuw Arch. Wisk. 6, 203-210, 1988.​
    but Nieuw Archief voor Wiskunde's online archives only go back to 2000.
     
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