MathDB

For any positive integer

k

denote by

S(k)

the number of solutions

(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+

of the system

\begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil \\ \mid x-y\mid =k , \end{cases}

where

d

is the greatest common divisor of positive integers

x

and

y.

Determine

S(k)

as a function of

k

. (Here

\lceil z\rceil

denotes the smalles integer number which is bigger or equal than

z.

)

Problem Statement