Big Perfect Square

Algebra Level 3

Find the smallest positive integer $x$ such that $\sqrt { { 4 }^{ x }+{ 4 }^{ 555 }+{ 4 }^{ 666 } }$ is a positive integer.

The answer is 443.

3 solutions

Andrew Ellinor
Oct 9, 2015

We want $4^x + 4^{555} + 4^{666}$ to take on a perfect square form, so we opt to rewrite it as $2^{2x} + 2*2^{1109} + 2^{1332}$ . If we try a factorization of $(2^x + 2^{666})(2^x + 2^{666})$ , then $x = 443$ .

Porames Vattanaprasan
Aug 22, 2015

In fact:

$\sqrt{4^{443}+4^{555}+4^{666}}=2^{443}\sqrt{1+4^{112}+4^{223}}$

$\sqrt{1+4^{112}+4^{223}}=\sqrt{1+2^{224}+4^{223}}=\sqrt{1+2\times2^{223}+(2^{2})^{223}}$ $=1+2^{223}$

So, the expression equals $2^{443}+2^{666}$ at $\boxed{x=443}$ which means the answer x would satisfy $x\leq443$ .

Gian Sanjaya - 5 years, 9 months ago

By the way, with wolframalpha, I found that:

$\sqrt{4^{666}+4^{555}}-(2^{666}+2^{443}-2^{219}+2^{-4})\approx-5.7956\times10^{-69}$

New edit:

Also, by wolframalpha help, I found that $\boxed{334\leq x \leq443}$

Gian Sanjaya - 5 years, 9 months ago

How can you know it is 443 if you don't have calculator or wolframalpha?

Porames Vattanaprasan - 5 years, 9 months ago

@Porames Vattanaprasan – I just stated x<=443, it could be lower but if you ask how 443 work, see above. I posted one solution.

Gian Sanjaya - 5 years, 9 months ago

@Gian Sanjaya – Good luck my friend :)

Porames Vattanaprasan - 5 years, 9 months ago

Magic Math
Aug 25, 2015

EASY PEASY LEMON SQUEEZY

Big Perfect Square

The answer is 443.

3 solutions

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