Calculator Fun

Number Theory Level 4

Once, I was playing with my calculator at school during the break. I was doing many things with the number 16, when I decided to calculate the square root of it. The result, of course, was 4. However, I pressed, by accident, the square root button again, and the result that was being shown was 2. I then thought of what had just happened: I had a number, then I calculated the square root of it, and the square root of the result was an integer, which was not a perfect square. In other words, the result of the square root of a perfect square (16) was also a perfect square (4). Let the initial number be $x$ , and the perfect square results be $x_{1}, x_{2}, x_{3}, ... ,x_{m}$ , where $x_{m}$ is the last possible result, which is not a perfect square. As an explicit example, I used $x = 16$ that day at school, so $m$ was $2$ and $x_{m} = 2$ . Consider that $n$ and $k$ are positive integers. In order to maximize the value of $m$ , $x$ should be of the form:

Note : the exponent of $x_m$ must be minimized.

Image credit: Like Cool.

n^{2^{k}}

n^{(2n)^{k}}

n^{k^{2}}

n^{2k}

kn^{2^{k}}

1 solution

Bruno Oggioni Moura
Sep 29, 2015

Suppose $x$ is of the form $n^y$ , where $y$ is a positive integer. Every time the square root of $x$ is calculated (remember that $x$ is a perfect square), we get as result $n^{\frac{y}{2}}$ . Doing it successively, we get a final result $x_{m}$ that can be expressed as $n^{y (\frac{1}{2})^{m}}$ . If we want to maximize the value of $m$ , while keeping the exponent even, the value of the exponent must be a potential of 2, so that it will always be divisible by 2, until it reaches $\frac{2}{2} = 1$ . So, $y = 2^{k}$ , where $k$ is a positive integer. Hence, $x$ must be of the form $\boxed{n^{2^{k}}}$

n^(2n)^k = n^(2^k * n^k) should be the answer , if the power can be divided m times by 2, then the value of m should be equal or higher than value of m of n^2^k as both contains 2^k as product in power. As the first one contains n^k also as product in power, m should be equal or higher than ur answer

Parker Prathyoom - 5 years, 8 months ago

You are definitely right. I'm really sorry about that. I edited the problem so that there won't be any other misunderstanding. Thanks for the comment!

Bruno Oggioni Moura - 5 years, 8 months ago

Calculator Fun

Image credit: Like Cool.

1 solution

0 pending reports