MathDB

Given a fixed rational number

q

. Let's call a number

x

charismatic if we can find a natural number

n

and integers

a_1, a_2,.., a_n

such that

x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .

i) Prove that one can find a

q

such that all positive rational numbers are charismatic. ii) Is it true that for all

q

, if the number

x

is charismatic, then

x + 1

is also charismatic?

Problem Statement