Question Number 150966 by mathdanisur last updated on 17/Aug/21
$$\mathrm{if}\:\:\mathrm{a};\mathrm{b}\in\mathbb{N}^{+} \:\:\mathrm{then}\:\mathrm{determine}\:\mathrm{all}\:\mathrm{the} \\ $$$$\mathrm{prime}\:\mathrm{numbers}\:\boldsymbol{\mathrm{p}}\:\mathrm{which}\:\mathrm{satisfy} \\ $$$$\left(\mathrm{p}\:+\:\mathrm{2}\right)^{\boldsymbol{\mathrm{a}}} \:=\:\left(\mathrm{p}\:-\:\mathrm{2}\right)^{\boldsymbol{\mathrm{b}}} \\ $$
Commented by Rasheed.Sindhi last updated on 17/Aug/21
$$\mathcal{S}{pecial}\:\mathcal{C}{ase}:\mathrm{a}=\mathrm{b} \\ $$$$\left(\mathrm{p}\:+\:\mathrm{2}\right)^{\boldsymbol{\mathrm{a}}} \:=\:\left(\mathrm{p}\:-\:\mathrm{2}\right)^{\boldsymbol{\mathrm{b}}} \\ $$$$\left(\mathrm{p}\:+\:\mathrm{2}\right)^{\boldsymbol{\mathrm{a}}} \:=\:\left(\mathrm{p}\:-\:\mathrm{2}\right)^{\mathrm{a}} \\ $$$$\mathrm{p}+\mathrm{2}=\mathrm{p}−\mathrm{2} \\ $$$$\mathrm{2}=−\mathrm{2}\:\mathrm{false} \\ $$$$\therefore\:\mathrm{a}\neq\mathrm{b} \\ $$
Commented by mathdanisur last updated on 17/Aug/21
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser} \\ $$$$\mathrm{The}\:\mathrm{equation}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{have}\:\mathrm{solution} \\ $$