Find all natural numbers n for which 2^n<=n^2

Answer 1

2^n = n^2

n= 2 or 4

2^2 = 2^2 = 4

or

2^4 = 4^2 = 16

there's a 3rd solution, but it's not an integer

a little more than -1 maybe about -3/4

one method is just try out a few simple small integers, such as 1,2,3 and 4

and Bingo, you discover 2 and 4 as solutions

Answer 2

I was writing up a whole answer, but I may have been overthinking this. Because they tell us they want natural numbers (0, 1, 2, 3...) you can just pic numbers to plug in. Also, because 2ⁿ can get really big really fast, and they want 2ⁿ to be less than(or equal to) n², we should pick small numbers to ensure it stays small.

For example, when n = 1, 2ⁿ = 2, and n² = 1 so 1 would not be a valid value for n.

when n = 2, 2ⁿ = 4, and n² = 4, which means this fulfills the requirements of the question.

I checked n=3 and n=4 which worked as well, but n=5 did not.

Therefore the answer is 2 <= n <= 4

This is not an easy problem, so if you would like me to explain any of the steps please feel free to reach out and ask. Hope this helped :)