if-a-b-N-then-determine-all-the-prime-numbers-p-which-satisfy-p-2-a-p-2-b-

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

Special Case:a=b (p + 2)^a = (p - 2)^b (p + 2)^a = (p - 2)^a p+2=p−2 2=−2 false ∴ a≠b

$$\mathcal{S}{pecial}\:\mathcal{C}{ase}:\mathrm{a}=\mathrm{b} \\ $$$$\left(\mathrm{p}\:+\:\mathrm{2}\right)^{\boldsymbol{\mathrm{a}}} \:=\:\left(\mathrm{p}\:-\:\mathrm{2}\right)^{\boldsymbol{\mathrm{b}}} \\ $$$$\left(\mathrm{p}\:+\:\mathrm{2}\right)^{\boldsymbol{\mathrm{a}}} \:=\:\left(\mathrm{p}\:-\:\mathrm{2}\right)^{\mathrm{a}} \\ $$$$\mathrm{p}+\mathrm{2}=\mathrm{p}−\mathrm{2} \\ $$$$\mathrm{2}=−\mathrm{2}\:\mathrm{false} \\ $$$$\therefore\:\mathrm{a}\neq\mathrm{b} \\ $$

Commented by mathdanisur last updated on 17/Aug/21

Thank you Ser The equation doesn′t have solution

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser} \\ $$$$\mathrm{The}\:\mathrm{equation}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{have}\:\mathrm{solution} \\ $$

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