Most problems of the additive number theory, with which we are dealing with in this chapter, are often hard to solve just. This is because they are additive in nature. The problem of finding numbers being the product of their divisors is related to the additive problem of finding perfect numbers, but much more simple to solve. We provide it here as an illustrative example of how much simpler a related multiplicative problem can be in relation to fining the solutions of an additive number-theoretic problem.

Proposition: Numbers Being the Product of Their Divisors

The natural number $n$ is the product of all its divisors $$\prod_{d\mid n}d=n^2$$ if and only if $n=p^3$ for some prime number $p$ or $n=p_ip_j$ for some prime numbers $p_i\neq p_j.$

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927