Problem
NT-B2-M06-P019 Exact Criterion
#19
★★★★★ Level 5 of 5
Let \(p\) be an odd prime, \(p\mid a-b\), \(p^2\nmid a-b\), and \(p\nmid ab\). Prove that \(p^r\mid a^n-b^n\) if and only if \(p^{r-1}\mid n\).
Under the conditions, \(v_p(a-b)=1\).
By LTE, \(v_p(a^n-b^n)=v_p(a-b)+v_p(n)=1+v_p(n)\). The divisibility \(p^r\mid a^n-b^n\) is equivalent to \(1+v_p(n)\ge r\), i.e. \(v_p(n)\ge r-1\). This is equivalent to \(p^{r-1}\mid n\).
Clean problem turning LTE into an iff criterion.