The section *Solved problems* from the Wikipedia *Floor and ceiling functions* shows several problems proposed by Ramanujan ([1]). The purpose of this post, if possible, is try to get the generalization of some of these identities, for positive integers $n\geq 1$, involving fractions or radicals and the floor function $\lfloor x\rfloor$.

I tried to get a generalization of the identity $(iii)$. I don't know if my conjectural identiy is in the literature or has a good mathematical content, these are my previous failed attempts.

**Counterexamples for different formulas.**

A counterexample for the (false) identity $$\lfloor \sqrt[k]{n}+\sqrt[k]{n+1}\rfloor=\lfloor \sqrt[k]{2^k n+k}\rfloor$$ is the integer $n=525$ for the case $k=5$.

Counterexamples for the (false) identity $$\lfloor \sqrt[k]{n}+\sqrt[k]{n+1}\rfloor= \left\lfloor \sqrt[k]{2^k n+2(k-1)} \right\rfloor$$ are the integers $n=11$ or $n=610$ for the case $k=6$.

From this thread of experiments I get the following conjecture.

**Conjecture.** *For each integer* $k\geq 2$ *one has that the identity*
$$\lfloor \sqrt[k]{n}+\sqrt[k]{n+1}\rfloor=\left\lfloor 2\sqrt[k]{n+\frac{1}{2}}\right\rfloor$$
*holds over integers* $n\geq 1$.

I don't know if it is easy to prove, or if you can to find a counterexample.

Question.Do you know if generalizations (thus with a good mathematical meaning, with mathematical significance) of the mentioned problems proposed by Ramanujan are in the literature? In this case, please refer the literature and I try to find and read from the literature these generalizations of $(i)$, $(ii)$ or $(iii)$. In case that aren't in the literature please add yourself generalization, if possible, with its respective proof for some of those identities. In particular, if you know that my conjeture can be proved or can be refuted by finding a counterexample.Many thanks.

Please if some professor/user finds a counterexample it is welcome that he/she comment it, many thanks. I add for example the following scripts written in Pari/GP as a proof of concept/toy model of my conjecture

`for(k=2,10,for(n=1, 1000,if(floor((n)^(1/k)+(n+1)^(1/k))!=floor(2*((n+1/2)^(1/k))),print(k," ",n))))`

that you can to evaluate from the web *Sage Cell Server*, just choose *GP* as language. Thus aren't showed conunterexamples as outputs. Also you've this one

`for(k=2,10,for(n=1, 100,print(floor((n)^(1/k)+(n+1)^(1/k)))))`

or this

`for(k=2,10,for(n=1, 100,print(floor((n)^(1/k)+(n+1)^(1/k))-floor(2*((n+1/2)^(1/k))))))`

I add as reference the PARI/GP Developers group of Université Bordeaux 1.

## References:

I believe that the corresponding reference is

[1] Srinivasa Ramanujan, *Collected Papers*, *Question 723* in p. 332, Providence RI: AMS / Chelsea (2000).

[2] I've used also the *PARI-GP Reference Card (version 2.2.5)*, by Karim Belabas (2003), based on an earlier version by Joseph H. Silverman.