# 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?

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#### shmoe

Homework Helper
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?

Last edited by a moderator:

#### Tide

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

#### shmoe

Homework Helper
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

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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