MathDB

Let

\nu

be an irrational positive number, and let

m

be a positive integer. A pair of

(a,b)

of positive integers is called good if

a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.

A good pair

(a,b)

is called excellent if neither of the pair

(a-b,b)

and

(a,b-a)

is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of

m

N7