Problem

Source: Iranian TST 2018, second exam day 2, problem 4

Tags: number theory, Iran, Iranian TST, TST



Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$. Prove that there exist infinitely many positive integers which they are not "useful but not optimized". (e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number) Proposed by Mohsen Jamali