Maximum positive integer that adds up to a perfect square?

[gundu]

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?

 

[shmoe]

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

What have you tried so far? Can you give any value of x that makes this a perfect square?

 

[HallsofIvy]

Assuming that x> 27,
$$4^27+ 4^1000+ 4^x= (4^{27})(1+ 4^{983}+ 4^{x- 27})$$

$$4^{27}= (4^{26})(2)$$
and
$$1+ 4^{983}+ 4^{x- 27}$$
is an odd number. What does that tell you?

 

[Tide]

I think Halls meant $4^{27} = 4^{26} \times 2^2$.

 

[shmoe]

Halls also means:

$$4^{27}+ 4^{1000}+ 4^x= (4^{27})(1+ 4^{973}+ 4^{x- 27})$$

(1000-27=973)

 

[vaishakh]

But the problem is to prove nothing is possible after that, hall.
Anyway gundu has to first clear what he has done as shmoesaid.

 

[HallsofIvy]

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:

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!