Maximum positive integer that adds up to a perfect square?

1. Dec 5, 2005

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?

2. Dec 6, 2005

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?

3. Dec 7, 2005

HallsofIvy

Staff Emeritus
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?

Last edited: Dec 7, 2005
4. Dec 7, 2005

Tide

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

5. Dec 7, 2005

shmoe

Halls also means:

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

(1000-27=973)

6. Dec 7, 2005

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.

7. Dec 8, 2005

HallsofIvy

Staff Emeritus
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!