Category: Number Theory

Question Number 9069 by tawakalitu last updated on 16/Nov/16 Commented by RasheedSoomro last updated on 20/Nov/16 $$\{\begin{array}{l}\mathrm{x}+\mathrm{y}=2\\ \mathrm{xy}=4\\ {\mathrm{S}}_{\mathrm{n}}={\mathrm{x}}^{\mathrm{n}}+{\mathrm{y}}^{\mathrm{n}}\end{array}$$$$\mathrm{pS}_{\mathrm{n}} =\mathrm{S}_{\mathrm{n}+\mathrm{1}} +\mathrm{qS}_{\mathrm{n}−\mathrm{1}} \…

Question Number 9025 by Rasheed Soomro last updated on 15/Nov/16 $$\mathrm{Determine}\mathrm{number}/\mathrm{s}\mathrm{that}\mathrm{is}/\mathrm{are}\mathrm{comprised}$$$$\mathrm{of}\mathrm{four}\mathrm{distinct}\mathrm{prime}\mathrm{factors}\mathrm{such}\mathrm{that}$$$$\mathrm{difference}\mathrm{of}\mathrm{largest}\mathrm{and}\mathrm{smallest}\mathrm{prime}$$$$\mathrm{factors}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{remaining}$$$$\mathrm{two}\mathrm{factors}.{}_{\mathrm{Propsed}\mathrm{by}\mathrm{Rasheed}\mathrm{Soomro}}$$ Commented by FilupSmith last…

Question Number 8838 by tawakalitu last updated on 31/Oct/16 $$\mathrm{Show}\mathrm{that}:{\mathrm{e}}^{\mathrm{i}\pi +1}=0$$ Answered by FilupSmith last updated on 31/Oct/16 $$\mathrm{do}\mathrm{you}\mathrm{mean}{e}^{i\pi }+1=0$$$${e}^{{ix}} =\mathrm{cos}\left({x}\right)+{i}\mathrm{sin}\left({x}\right)…

Question Number 139560 by Dwaipayan Shikari last updated on 28/Apr/21 $$\underset{{n}_{\mathrm{1}} +\mathrm{2}{n}_{\mathrm{2}} +\mathrm{3}{n}_{\mathrm{3}} +..+{rn}_{{r}} ={n}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}_{\mathrm{1}} !{n}_{\mathrm{2}} !{n}_{\mathrm{3}} !..{n}_{{r}} !\mathrm{1}^{{n}_{\mathrm{1}} } \mathrm{2}^{{n}_{\mathrm{2}} } \mathrm{3}^{{n}_{\mathrm{3}}…

Question Number 8336 by Rasheed Soomro last updated on 08/Oct/16 $$\mathrm{Determine}\mathrm{smallest}\mathrm{n}(\ne 0),\mathrm{for}\mathrm{which}$$$${(\omega +\mathrm{i})}^{\mathrm{n}}=1.$$ Commented by prakash jain last updated on 08/Oct/16 $${w}=\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{3}}=−\frac{\mathrm{1}}{\mathrm{2}}+{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}…