The section *problem solved* from Wikipedia *Floor and ceiling functions* shows several problems proposed by Ramanujan ((1)). The purpose of this post, if possible, is to generalize some of these identities to positive integers $ n geq 1 $involving fractions or radicals and soil function $ lfloor x rfloor $,

I tried to generalize the identity $ (iii) $, I do not know if my guess identity is in the literature or has good mathematical content. These are my previous failed attempts.

**Counterexamples for different formulas.**

1) A counterexample of (false) identity

$$ lfloor sqrt (k) {n} + sqrt (k) {n + 1} rfloor = lfloor sqrt (k) {2 ^ k n + k} rfloor $$

is the whole number $ n = $ 525 In the event of $ k = 5 $,

2) counterexamples for the (false) identity

$$ lfloor sqrt (k) {n} + sqrt (k) {n + 1} rfloor = lfloor sqrt (k) {2 ^ k n + 2 (k-1)} rfloor $$

are the integers $ n = 11 $ or $ n = 610 $ In the event of $ k = 6 $,

From this thread of experiments, I get the following guess.

**Guess.** *For every integer* $ k geq $ 2 *it has the identity*

$$ lfloor sqrt (k) {n} + sqrt (k) {n + 1} rfloor = lfloor 2 sqrt (k) {n + frac {1} {2}} rfloor $$

*holds over integers* $ n geq 1 $,

I do not know if it's easy to prove or if you can find a counterexample.

**Question.** Do you know whether generalizations (that is, with a good mathematical meaning, with mathematical significance) of the problems suggested by Ramanujan are present in the literature? In this case, please refer to the literature and I try to find and read these generalizations from the literature $ (i) $. $ (ii) $ or $ (iii) $, If this is not in the literature, please add, if possible, a generalization with proof of some of these identities. Especially if you know that my incantation can be proved or disproved by finding a counterexample. **Many thanks.**

## references:

I believe that is the corresponding reference

(1) Srinivasa Ramanujan, *Collected papers*. *Inquiry 723* in p. 332, Providence RI: AMS / Chelsea (2000).