Question Number 121870 by mnjuly1970 last updated on 12/Nov/20
$$\:\:\:\:\:\:{number}\:\:{theory}: \\ $$$$\:\:\:\:\:\:{m},{n}\:\in\:\mathbb{N}\:\:,\:\:\left({m},{n}\right)=\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\::\:\:\:{m}^{\varphi\left({n}\right)} +{n}^{\varphi\left({m}\right)} \overset{{mn}} {\equiv}\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\varphi\left({n}\right)=\mid\left\{{x}\in\mathbb{N}\mid\:{x}<{n}\:,\:\left({x},{n}\right)=\mathrm{1}\right\}\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}. \\ $$
Answered by mindispower last updated on 12/Nov/20
![m^(ϕ(n)) =1[n]....euler theorm t n^(ϕ(m)) =1[m] ⇒n∣m^(ϕ(n)) −1,m∣n^(ϕ(m)) −1 ⇒nm∣(m^(ϕ(n)) −1)(n^(ϕ(m)) −1) ⇔nm∣(m^(ϕ(n)) .n^(ϕ(m)) −(n^(ϕ(m)) +m^(ϕ(n)) −1)) ⇔mn∣mn(m^(ϕ(n)−1) .n^(ϕ(m)−1) −(n^(ϕ(m)) +m^(ϕ(n)) −1) ⇔mn∣−(n^(ϕ(m)) +m^(ϕ(n)) −1) ⇒n^(ϕ(m)) +m^(ϕ(n)) ≡1(mn)](https://www.tinkutara.com/question/Q121940.png)
$${m}^{\varphi\left({n}\right)} =\mathrm{1}\left[{n}\right]….{euler}\:{theorm}\:{t} \\ $$$${n}^{\varphi\left({m}\right)} =\mathrm{1}\left[{m}\right] \\ $$$$\Rightarrow{n}\mid{m}^{\varphi\left({n}\right)} −\mathrm{1},{m}\mid{n}^{\varphi\left({m}\right)} −\mathrm{1} \\ $$$$\Rightarrow{nm}\mid\left({m}^{\varphi\left({n}\right)} −\mathrm{1}\right)\left({n}^{\varphi\left({m}\right)} −\mathrm{1}\right) \\ $$$$\Leftrightarrow{nm}\mid\left({m}^{\varphi\left({n}\right)} .{n}^{\varphi\left({m}\right)} −\left({n}^{\varphi\left({m}\right)} +{m}^{\varphi\left({n}\right)} −\mathrm{1}\right)\right) \\ $$$$\Leftrightarrow{mn}\mid{mn}\left({m}^{\varphi\left({n}\right)−\mathrm{1}} .{n}^{\varphi\left({m}\right)−\mathrm{1}} −\left({n}^{\varphi\left({m}\right)} +{m}^{\varphi\left({n}\right)} −\mathrm{1}\right)\:\right. \\ $$$$\Leftrightarrow{mn}\mid−\left({n}^{\varphi\left({m}\right)} +{m}^{\varphi\left({n}\right)} −\mathrm{1}\right) \\ $$$$\Rightarrow{n}^{\varphi\left({m}\right)} +{m}^{\varphi\left({n}\right)} \equiv\mathrm{1}\left({mn}\right) \\ $$
Commented by mnjuly1970 last updated on 13/Nov/20
$${good}\:\:{very}\:{good}\:{mr}\:{power}.. \\ $$$${thank}\:{you}.. \\ $$