Question Number 34585 by tawa tawa last updated on 08/May/18
Commented by Rasheed.Sindhi last updated on 08/May/18
$$\mathcal{N}{ot}\:{related}\:{to}\:{above}…. \\ $$$$\mathrm{Q}#\mathrm{34429}\:\mathrm{by}\:\mathrm{S}@\mathrm{ty}@\mathrm{m}:\:\mathrm{3}^{\mathrm{m}} +\mathrm{3}^{\mathrm{n}} +\mathrm{1}\:\mathrm{is}\:\mathrm{not} \\ $$$$\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}.\:\mathrm{I}'\mathrm{ve}\:\mathrm{tried}\:\mathrm{to}\:\mathrm{provide} \\ $$$$\mathrm{a}\:\mathrm{proof}.\:\mathrm{Anyone}\:\mathrm{who}\:\mathrm{is}\:\mathrm{interested}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{question}\:\mathrm{can}\:\mathrm{see}\:\mathrm{my}\:\mathrm{proof}.\mathrm{Critical} \\ $$$$\mathrm{comments}\:\mathrm{will}\:\mathrm{be}\:\mathrm{welcomed}. \\ $$
Answered by MJS last updated on 09/May/18
$${r}_{\mathrm{0}} =\mathrm{4} \\ $$$${M}_{\mathrm{0}} {M}_{\mathrm{1}} ={M}_{\mathrm{0}} {M}_{\mathrm{2}} ={r}_{\mathrm{1}} ={r}_{\mathrm{2}} =\mathrm{2} \\ $$$${M}_{\mathrm{0}} {M}_{\mathrm{3}} ={r}_{\mathrm{0}} −{r}_{\mathrm{3}} \\ $$$${M}_{\mathrm{1}} {M}_{\mathrm{3}} ={M}_{\mathrm{2}} {M}_{\mathrm{3}} ={r}_{\mathrm{1}} +{r}_{\mathrm{3}} \\ $$$$\left({M}_{\mathrm{0}} {M}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({M}_{\mathrm{0}} {M}_{\mathrm{3}} \right)^{\mathrm{2}} =\left({M}_{\mathrm{1}} {M}_{\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$\mathrm{2}^{\mathrm{2}} +\left(\mathrm{4}−{r}_{\mathrm{3}} \right)^{\mathrm{2}} =\left(\mathrm{2}+{r}_{\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$\mathrm{4}+\mathrm{16}−\mathrm{8}{r}_{\mathrm{3}} +{r}_{\mathrm{3}} ^{\mathrm{2}} =\mathrm{4}+\mathrm{4}{r}_{\mathrm{3}} +{r}_{\mathrm{3}} ^{\mathrm{2}} \\ $$$$\mathrm{16}=\mathrm{12}{r}_{\mathrm{3}} \\ $$$${r}_{\mathrm{3}} =\frac{\mathrm{4}}{\mathrm{3}} \\ $$
Commented by Joel578 last updated on 09/May/18
$${what}\:{is}\:{M}\:? \\ $$
Commented by MJS last updated on 09/May/18
$$\mathrm{the}\:\mathrm{centers}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circles}.\:\mathrm{sorry},\:\mathrm{my}\:\mathrm{language} \\ $$$$\mathrm{is}\:\mathrm{German}\:\mathrm{center}\:\mathrm{is}\:“\mathrm{Mittelpunkt}''\:\mathrm{in}\:\mathrm{German} \\ $$$$\mathrm{so}\:\mathrm{without}\:\mathrm{further}\:\mathrm{thinking},\:{M}\:\mathrm{seemed}\:\mathrm{natural}… \\ $$
Commented by Joel578 last updated on 10/May/18
$${okay},\:{thank}\:{you} \\ $$