Question

Prove that there exists a pair of consecutive integers such that one of these integers is a perfect square and the other is a perfect cube.

(I guessed the integers to prove this, but I wanted to see if there was another way rather than guessing the two integers and writing them down as my proof.)

Answer 1

You just have to show that such a pair indeed exists, so an example suffices. Did you use 0 and 1?

Answer 2

Other than that, I'm not sure how to (dis)prove this.

Answer 3

I used 8 and 9. 8 is 2^3 and 9 is 3^2

Answer 4

Maybe guessing the example is just the way you go about proving it..

Answer 5

Yeah, the problem only wants one. That's proof enough for this kind of problem.

Answer 6

Meh, it just feels odd that this is the way to prove the problem. By guessing and writing down the example... But whatever, if that's all there is to it then so be it.

Answer 7

how about you with the numbers :
4^3 and 3^4 = (3^2)^2, is it allowed ?