MathDB

Does there exist a nonnegative integer

a

for which the equation

\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a

has more than one million different solutions

(m, n)

where

m

and

n

are positive integers?
The expression

\lfloor x\rfloor

denotes the integer part (or floor) of the real number

x

. Thus

\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,

and

\lfloor 0 \rfloor = 0

Problem Statement