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Maximum positive integer that adds up to a perfect square?

  1. Dec 5, 2005 #1
    4 to the power of 27 + 4 to the power of 1000 + 4 to the power of x.
    x is the maximum positive integer and it adds up to a perfect square?
     
  2. jcsd
  3. Dec 6, 2005 #2

    shmoe

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    To clarify your question, you are asking for the largest integer x such that [tex]4^{27}+4^{1000}+4^{x}[/tex] is a perfect square?

    What have you tried so far? Can you give any value of x that makes this a perfect square?
     
  4. Dec 7, 2005 #3

    HallsofIvy

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    Assuming that x> 27,
    [tex]4^27+ 4^1000+ 4^x= (4^{27})(1+ 4^{983}+ 4^{x- 27})[/tex]

    [tex]4^{27}= (4^{26})(2)[/tex]
    and
    [tex]1+ 4^{983}+ 4^{x- 27}[/tex]
    is an odd number. What does that tell you?
     
    Last edited: Dec 7, 2005
  5. Dec 7, 2005 #4

    Tide

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    I think Halls meant [itex]4^{27} = 4^{26} \times 2^2[/itex].
     
  6. Dec 7, 2005 #5

    shmoe

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    Halls also means:

    [tex]4^{27}+ 4^{1000}+ 4^x= (4^{27})(1+ 4^{973}+ 4^{x- 27})[/tex]

    (1000-27=973)
     
  7. Dec 7, 2005 #6
    But the problem is to prove nothing is possible after that, hall.
    Anyway gundu has to first clear what he has done as shmoesaid.
     
  8. Dec 8, 2005 #7

    HallsofIvy

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    No, the OP said:
    Which I interpret to mean "What is the largest positive integer such that this adds to a perfect square.

    Of course, since I clearly can't do basic arithmetic, I can't answer this!
     
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