Competition problem — Number theory

Library / /44 of 214

Number theory Difficulty 5.4 AIME, harder Prove it Belarus

Does there exist a positive integer $n$ which can be represented both as $n = a^2 - b$ and $n = b^2 - c$ , where $a$ , $b$ , $c$ are three distinct divisors of $n$ ?

Solution

Answer: no, such $n$ doesn't exist.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.