Category: Number Theory

Question Number 23105 by Rasheed.Sindhi last updated on 26/Oct/17 $$(\mid {\mathrm{m}}^2-{\mathrm{n}}^2\mid ,2\mathrm{mn},{\mathrm{m}}^2+{\mathrm{n}}^2)\mathrm{is}\mathrm{pythagorean}$$$$\mathrm{triplet}\mathrm{for}\mathrm{all}\mathrm{m},\mathrm{n}\in \mathbb{N}.\mathrm{This}\mathrm{can}\mathrm{be}$$$$\mathrm{proved}\mathrm{easily}.\mathrm{Is}\mathrm{the}\mathrm{vice}\mathrm{versa}\mathrm{of}$$$$\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{also}\mathrm{true}?$$$$\mathrm{I}-\mathrm{E}$$$$\mathrm{If}\:\:\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N}\:,\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}}…

Question Number 88261 by ar247 last updated on 09/Apr/20 Answered by Joel578 last updated on 09/Apr/20 $$u\equiv v\left(\mathrm{mod}m\right)\Rightarrow u-v=mx,x\in \mathbb{Z}$$$$\left(1\right)u=mx+v$$$$\mathrm{Since}\mathrm{gcf}(v,m)\mathrm{divides}\mathrm{both}v\mathrm{and}\mathrm{m},\mathrm{it}\mathrm{also}\mathrm{divides}u,$$$$\mathrm{hence}\mathrm{gcf}(v,m)\mathrm{divides}\mathrm{gcf}(u,m)$$$$\Rightarrow\:\mathrm{gcf}\left({v},{m}\right)\:\mid\:\mathrm{gcf}\left({u},{m}\right)…

Question Number 22625 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\mathrm{each}\mathrm{positive}\mathrm{integer}\mathrm{n}\mathrm{define}$$$${\mathrm{a}}_{\mathrm{n}}=30+{\mathrm{n}}^2,\mathrm{and}{\mathrm{d}}_{\mathrm{n}}=\gcd ({\mathrm{a}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}+1}).$$$$\mathrm{Find}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{values}\mathrm{that}\mathrm{are}$$$$\mathrm{taken}\mathrm{by}{\mathrm{d}}_{\mathrm{n}}\mathrm{and}\mathrm{show}\mathrm{by}\mathrm{examples}$$$$\mathrm{that}\:\mathrm{each}\:\mathrm{of}\:\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \…

Question Number 88000 by jdmath last updated on 07/Apr/20 $$Isthereaformulatocalculate$$$$$$$$\underset{i=1}{\sum ^n}\frac{1}{i^2}$$$$intermsofn..?$$ Commented by abdomathmax last…

Question Number 22372 by Rasheed.Sindhi last updated on 16/Oct/17 Commented by Rasheed.Sindhi last updated on 16/Oct/17 $$\mathrm{Question}\mathrm{asked}\mathrm{by}\mathrm{Mr}.\mathrm{Tinkutara}.$$$$\mathrm{reposted}\mathrm{for}\mathrm{answer}.$$ Commented by Tinkutara last…