Let-x-and-y-be-integers-such-that-xy-1-x-2-y-and-y-2-x-i-Show-that-p-xy-1-and-p-x-2-y-then-p-y-2-x-where-p-is-a-prime-ii-Let-p-be-a-prime-Suppose-that-p-x-2-y-and-p-y-2-x-must-p-xy-1-

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Question Number 110562 by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{Let}\mathrm{x}\mathrm{and}\mathrm{y}\mathrm{be}\mathrm{integers}\mathrm{such}\mathrm{that}$$$$\mathrm{xy}\ne 1,{\mathrm{x}}^2\ne \mathrm{y}\mathrm{and}{\mathrm{y}}^2\ne \mathrm{x}.$$$$$$$$\left(\mathrm{i}\right)\mathrm{Show}\mathrm{that}\mathrm{p}\mid \mathrm{xy}-1\mathrm{and}\mathrm{p}\mid {\mathrm{x}}^2-\mathrm{y}$$$$\mathrm{then}\mathrm{p}\mid {\mathrm{y}}^2-\mathrm{x}\mathrm{where}\mathrm{p}\mathrm{is}\mathrm{a}\mathrm{prime}.$$$$\left(\mathrm{ii}\right)\mathrm{Let}\mathrm{p}\mathrm{be}\mathrm{a}\mathrm{prime}.\mathrm{Suppose}\mathrm{that}$$$$\mathrm{p}\mid {\mathrm{x}}^2-\mathrm{y}\mathrm{and}\mathrm{p}\mid {\mathrm{y}}^2-\mathrm{x},\mathrm{must}\mathrm{p}\mid \mathrm{xy}-1?$$$$$$$$[\mathrm{If}\mathrm{yes},\mathrm{then}\mathrm{prove}\mathrm{it}.\mathrm{If}\mathrm{no},\mathrm{then}\mathrm{give}\mathrm{a}$$$$\mathrm{counter}\mathrm{example}]$$

Answered by Aziztisffola last updated on 29/Aug/20

$$\left(\mathrm{i}\right)\mathrm{p}\mid \mathrm{xy}-1\Rightarrow \mathrm{p}\mid \mathrm{x}(\mathrm{xy}-1)\Rightarrow \mathrm{p}\mid {\mathrm{x}}^2\mathrm{y}-\mathrm{x}$$$$\mathrm{p}\mid {\mathrm{x}}^2-\mathrm{y}\Rightarrow \mathrm{p}\mid {\mathrm{yx}}^2-{\mathrm{y}}^2$$$$\Rightarrow \mathrm{p}\mid {\mathrm{x}}^2\mathrm{y}-\mathrm{x}-{\mathrm{yx}}^2+{\mathrm{y}}^2\Rightarrow \mathrm{p}\mid {\mathrm{y}}^2-\mathrm{x}$$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{Thanks}\mathrm{but}\mathrm{why}\mathrm{does}\mathrm{p}\mid \mathrm{x}(\mathrm{xy}-1)?$$

Commented by Aziztisffola last updated on 29/Aug/20

$$\mathrm{p}\mid (\mathrm{xy}-1)\Rightarrow \mathrm{xy}-1\equiv 0\left[\mathrm{p}\right]$$$$\Rightarrow \mathrm{x}(\mathrm{xy}-1)\equiv 0\left[\mathrm{p}\right]\Rightarrow \mathrm{p}\mid \mathrm{x}(\mathrm{xy}-1)$$

Answered by Aziztisffola last updated on 29/Aug/20

$$\left(\mathrm{ii}\right)\mathrm{No}$$$$\mathrm{p}=3\mathrm{and}\mathrm{x}=\mathrm{y}=3$$$$3\mid 3^2-3\mathrm{and}3\mid 3^2-3\nRightarrow 3\nmid 8$$$$\mathrm{or}\mathrm{p}=2;\mathrm{x}=8\mathrm{and}\mathrm{y}=4$$

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