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Question Number 163845 by mnjuly1970 last updated on 12/Jan/22

$$$$$$a,b\in \mathbb{N},gcd(a,b)=1$$$$provethat$$$$$$$$a^{\phi \left(b\right)}+b^{\phi \left(a\right)}\stackrel{ab}{\equiv }1$$$$\phi :\mathcal{E}ulerphifunction...$$$$$$

Answered by mindispower last updated on 14/Jan/22

$$a^{\phi \left(b\right)}\equiv 1\left[b\right]\iff a^{\phi \left(b\right)}-1\equiv 0\left[b\right]$$$$b^{\phi \left(a\right)}\equiv 1\left[a\right]\iff b^{\phi \left(a\right)}-1\equiv 0\left[a\right]..fermatTheorem$$$$\Rightarrow (a^{\phi \left(b\right)}-1)(b^{\phi \left(a\right)}-1)\equiv 0\left[ab\right]$$$$\iff a^{\phi \left(b\right)}{b}^{\phi \left(a\right)}-(a^{\phi \left(b\right)}+b^{\phi \left(a\right)}-1)\equiv 0\left[ab\right].....E$$$$\phi :{\mathbb{N}}^{\ast }\to {\mathbb{N}}^{\ast }\Rightarrow \phi \left(a\right),\phi \left(b\right)⩾1\Rightarrow ab\mid a^{\phi \left(b\right)}{b}^{\phi \left(a\right)}$$$$a^{\phi \left(b\right)}{b}^{\phi \left(a\right)}\equiv 0\left[ab\right]$$$$E\iff b^{\phi \left(a\right)}+a^{\phi \left(b\right)}-1\equiv \left[ab\right]\iff b^{\phi \left(a\right)}+a^{\phi \left(b\right)}\equiv 1\left[b\right]$$$$$$

Commented by mnjuly1970 last updated on 14/Jan/22

$$thanksalotsirpower..$$

Commented by mindispower last updated on 18/Jan/22

$$withePleasursir$$

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