Number of Perfect Squares

Number Theory Level 3

$\Big\{1^1,\ 2^2,\ 3^3,\ \ldots,\ 2017^{2017}\Big\}$

How many perfect squares are there in the above set?

The answer is 1030.

1 solution

Steven Yuan
Jun 13, 2017

A number of the form $x^x,$ where $x$ is some positive integer, can be a perfect square if and only if either $x$ is even or $x$ is an odd perfect square. There are 1008 even integers between 1 and 2017 inclusive. To count the number of odd perfect squares in that interval, we realize that $44^2$ is the greatest perfect square less than 2017. There are 22 odd integers between 1 and 44 inclusive, so there are also 22 odd perfect squares between 1 and 2017. Therefore, the number of perfect squares in the set is $1008 + 22 = \boxed{1030}.$

I do not understand how 44^2=1936 will be included in the Odd Perfect squares, since 44 is an even number and the number is already included in the 1008 even integers.

Sivasis Dash - 3 years, 11 months ago

I used $44^2$ to give an upper bound for the value of an odd perfect square less than 2017, but of course that's not an odd perfect square - it's only an upper bound.

Steven Yuan - 3 years, 11 months ago

Number of Perfect Squares

The answer is 1030.

1 solution

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