Elementary Number Theory

Problem 1

Problem Find all $n\in \mathbb{N}\setminus \{0\}$ such that $(n+1)|(n^2+1)$ Solution $\stackrel{def}{⟹}\exists x(x(n+1)=n^2+1)$ There are three obvious cases:

$$\{\begin{array}{l}x\le n-1⟹x(n+1)\le n^2-1<n^2+1\text{ contradiction}\\ x=n⟹x(n+1)=n^2+n=n^2+1⟹n=1\\ x\ge n+1⟹x(n+1)\ge n^2+2n+1>n^2+1\text{ contradiction}\end{array}$$

This technique can be used in finding many integer equation systems (Diophantine equations).