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Maximum positive integer that adds up to a perfect square?

  • Thread starter gundu
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  • #1
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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?
 

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  • #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?
 
  • #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?
 
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  • #4
Tide
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I think Halls meant [itex]4^{27} = 4^{26} \times 2^2[/itex].
 
  • #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)
 
  • #6
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But the problem is to prove nothing is possible after that, hall.
Anyway gundu has to first clear what he has done as shmoesaid.
 
  • #7
HallsofIvy
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vaishakh said:
But the problem is to prove nothing is possible after that, hall.
Anyway gundu has to first clear what he has done as shmoesaid.
No, the OP said:
gundu said:
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?
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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