$m$ doesn't come in the succession $a_n=$ iff m is a perfect square (1 answer)


You are watching: The number -4 does not belong to which set of numbers

define which herbal numbers do not belong to the set $$E = \left\\left \mid n \in \tasiilaq.netbbN\right\.$$

The price is the set of hopeful perfect squares. I am not sure just how to prove it. We understand that $\left = n+\left<\sqrtn+\frac12 \right>$, but what carry out I execute from here?


*

Let $f(n):=n+\sqrtn+\frac12$ and also $g(n):=\big\lfloor f(n)\big\rfloor$ because that each $n=1,2,\ldots$. Observe the $f(n)$ is no an creature for any type of positive integer $n$. We have $1k+2\,.$$Hence, $$\left(\sqrtn+\frac12\right)^2k+\frac74\,.$$Therefore,$$\sqrtn\sqrtk+\frac74-\frac12\,.$$Ergo,$$k+1-\sqrtk+\frac74l^2+l+1+\sqrtl^2+l+\frac14+\frac12=l^2+2l+2\,.$$Therefore, $g\left(l^2+l\right)=l^2+2l=(l+1)^2-1$ and $g\left(l^2+l+1\right)=l^2+2l+2=(l+1)^2+1$. Hence, $(l+1)^2\notin E$ because that every $l=1,2,\ldots$. The proof is now complete.


*

when walk $\lfloor \sqrt n + \frac 12\rfloor$ jump?

$(k+0.5)^2 = k^2 + k + 0.25$

when $n$ can be composed as $k^2 + k$ you are on the low side that the next step, and also when $n$ deserve to be composed as $k^2 + k + 1$ you space on the high next of a step.

$\lfloor k^2 + k +\sqrt k^2 + k + \frac 12\rfloor = k^2 + 2k\\\lfloor k^2 + k + 1+\sqrt k^2 + k + 1 + \frac 12\rfloor = k^2 + 2k + 2\\$

And, $(k^2 + 2k + 1) = (k+1)^2$ gets skipped.


*

Not the answer you're spring for? Browse other questions tagged number-theory or ask your own question.


If each each the two herbal numbers $a$ and $b$ is a amount of two squares climate $ab$ is likewise a sum of two squares
uncover the quantity of natural numbers that deserve to be composed as $x^2$, $x^3$ and also $x^5$ that are smaller or equal 보다 $2^30$
*

site design / logo © 2021 stack Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.9.24.40305




See more: How To Wake Up Snorlax In Pokemon Soul Silver ? How To Wake Up Snorlax In Pokemon Soul Silver

tasiilaq.netematics stack Exchange works ideal with JavaScript allowed
*

her privacy

By clicking “Accept every cookies”, you agree ridge Exchange can store cookies on your machine and disclose info in accordance v our Cookie Policy.