How prove or disprove $\lfloor a^m\rfloor \neq \lfloor b^n\rfloor$ for all $m,n\in\mathbb{N}$

Question

Question:

prove or disprove: there exsit irrational $a>1,b>1$ such that for all positive integers $m,n$,

$$\lfloor a^m\rfloor \neq \lfloor b^n\rfloor$$

Now I can't prove this problem.

I know this following: $$a^m-\{a^m\}\neq b^n-\{b^n\}$$ since $$a^m-b^n\neq \{a^m\}-\{b^n\}\in (-1,2)$$

I only have this idea. if one can take example, Thank you

Answer 1

You may be interested to look at information on Beatty sequences and Rayleigh's Theorem: http://en.wikipedia.org/wiki/Beatty_sequence#Rayleigh_theorem