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Decimal integers with nonzero digits and sum of powers puzzle

  1. Feb 20, 2009 #1
    Determine all possible positive decimal integer(s) P = X1X2X3….Xn, where P>=2 with none of the digits in P being zero, that satisfy this equation:

    P = X1^X1 + X2^X2 + ……+ Xn^Xn

    (For example, P = 234 cannot be a solution since 2^2 + 3^3 + 4^4 is equal to 287, not 234.)


    (i) X1X2X3….Xn denotes the concatenation of the digits X1, X2, …, Xn and do not represent the product of the digits.

    (ii) P cannot admit any leading zero.
  2. jcsd
  3. Feb 20, 2009 #2
    I'm assuming that X1, X2, X3 are all the individual *digits* of P?

    Last edited: Feb 20, 2009
  4. Feb 20, 2009 #3
    Yes, each of X1,X2, ....,Xn correspond to an individual digit of P.
  5. Feb 20, 2009 #4
    Can any digit be the same as another digit?
  6. Feb 20, 2009 #5


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    ...or presumably, any other zeros.
  7. Feb 20, 2009 #6
    Here are a few:
    Of these only the first meets the unstated condition that all digits be distinct. The third one fails to meet the unstated condition that no digits be 0. I used a brute force method to get this partial solution. I do not know if this list is complete.
  8. Feb 23, 2009 #7
    That also assumes that 0^0 = 0-- I thought the more accepted solution was that 0^0 = 1?

    Otherwise, I think there's an upper bound of roughly 3.4 billion. Beyond that I think the rate at which the sum of the powers increases is capped (since 9^9 is the highest sum a digit can contribute), and the number itself is increasing more quickly.

  9. Feb 23, 2009 #8
    Good catch. There was a bug in my code. So I have only found 2 solutions.
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