Can you make all natural numbers using $x^2$ and $\frac{x}{2}$?

Asked May 20 '18 at 11:13

Active May 20 '18 at 13:44

Viewed 84 times

I came over a recent post that asked if you can make all natural numbers from the number 3 using factorials and square roots. This is a question in the same vein that seems more approachable, and which may give clues as how to solve the related problem.

All varibles used in the post are natural numbers.

Using these fuctions, is it possible to make all natural numbers from the number 3?

$$x^2,\; \frac{x}{2},\;\lceil x\rceil,\;\lfloor x \rfloor$$ Examples:

$1=\lfloor \frac{3}{2} \rfloor$

$2=\lceil \frac{3}{2} \rceil$

$3=3$

$4=\lfloor \frac{3^2}{2} \rfloor$

$5=\lceil \frac{3^2}{2} \rceil$

Notes:

The answer is yes if all natural numbers can be expressed as $\lfloor \frac{3^{2^a}}{2^b} \rfloor$ or $\lceil \frac{3^{2^a}}{2^b} \rceil$.

edited May 20 '18 at 13:44

asked May 20 '18 at 11:13

Ola

how will you create 10 – Abhishek Choudhary May 20 '18 at 11:19
@Abhishek $10=\lfloor (\frac{3^2}{2})^2 \rfloor$ – Ola May 20 '18 at 11:24
its not 10 it gives 12 – Abhishek Choudhary May 20 '18 at 11:30
@Abhishek It does give 10. See http://www.wolframalpha.com/input/?i=floor((((3%5E2)%2F2)%5E2)%2F2) – Ola May 20 '18 at 11:33
you forgot to write $\frac{1}{2}$ inside floor function – Abhishek Choudhary May 20 '18 at 11:39
I see. My misstake. – Ola May 20 '18 at 11:40
$\lfloor \frac{3^4}{2^3}\rfloor =10$ , hence $10$ can be created. – Peter May 20 '18 at 17:11

Can you make all natural numbers using $x^2$ and $\frac{x}{2}$?

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