Menu Close

Category: Number Theory

Let-us-define-the-positive-number-n-with-four-digits-a-b-c-and-d-such-that-n-abcd-with-a-b-c-d-Z-1-a-9-0-b-9-0-c-9-and-0-d-9-Let-us-then-say-that-a-cool-number-is-a-four-digit-number-say-n-such-

Question Number 4540 by Yozzii last updated on 06/Feb/16 $${Let}\:{us}\:{define}\:{the}\:{positive}\:{number}\:{n}\:{with}\:{four} \\ $$$${digits}\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}}\:{and}\:\boldsymbol{\mathrm{d}}\:{such}\:{that}\:{n}=\boldsymbol{\mathrm{abcd}} \\ $$$${with}\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}},\boldsymbol{\mathrm{d}}\in\mathbb{Z},\:\mathrm{1}\leqslant\boldsymbol{\mathrm{a}}\leqslant\mathrm{9},\:\mathrm{0}\leqslant\boldsymbol{\mathrm{b}}\leqslant\mathrm{9}, \\ $$$$\mathrm{0}\leqslant\boldsymbol{\mathrm{c}}\leqslant\mathrm{9}\:{and}\:\mathrm{0}\leqslant\boldsymbol{\mathrm{d}}\leqslant\mathrm{9}.\:{Let}\:{us}\:{then}\:{say} \\ $$$${that}\:{a}\:{cool}\:{number}\:{is}\:{a}\:{four}\:{digit}\:{number}, \\ $$$${say}\:{n},\:{such}\:{that}\:{the}\:{two}\:{digit}\:{numbers}\:{written}\:{as} \\ $$$$\boldsymbol{\mathrm{ab}}\:{and}\:\boldsymbol{\mathrm{cd}}\:{are}\:{given}\:{by}\:\boldsymbol{\mathrm{ab}}={r}×{s}\:{and} \\ $$$$\boldsymbol{\mathrm{cd}}=\left({r}−\mathrm{1}\right)×\left({s}+\mathrm{1}\right)\:{for}\:{some}\:{non}−{negative}\:{integers} \\…

Let-p-q-and-r-be-the-distinct-roots-of-the-polynomial-x-3-22x-2-80x-67-There-exist-real-number-A-B-and-C-such-that-1-s-3-22s-2-80s-67-A-s-p-B-s-q-C-s-r-for-all-real-numbers-

Question Number 135566 by bemath last updated on 14/Mar/21 $${Let}\:{p},{q}\:{and}\:{r}\:{be}\:{the}\:{distinct}\:{roots} \\ $$$${of}\:{the}\:{polynomial}\:{x}^{\mathrm{3}} −\mathrm{22}{x}^{\mathrm{2}} +\mathrm{80}{x}−\mathrm{67}. \\ $$$${There}\:{exist}\:{real}\:{number}\:{A},{B}\:{and} \\ $$$${C}\:{such}\:{that}\:\frac{\mathrm{1}}{{s}^{\mathrm{3}} −\mathrm{22}{s}^{\mathrm{2}} +\mathrm{80}{s}−\mathrm{67}}\:= \\ $$$$\frac{{A}}{{s}−{p}}\:+\:\frac{{B}}{{s}−{q}}\:+\:\frac{{C}}{{s}−{r}}\:{for}\:{all}\:{real}\:{numbers} \\ $$$${s}\:{with}\:{s}\:\notin\:\left\{{p},{q},{r}\right\}.{What}\:{is}\: \\…

1-1-cos-i-sin-i-1-

Question Number 135557 by bemath last updated on 14/Mar/21 $$\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\:\theta−{i}\:\mathrm{sin}\:\theta}\:=? \\ $$$${i}=\sqrt{−\mathrm{1}} \\ $$ Answered by mathmax by abdo last updated on 14/Mar/21 $$\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\theta−\mathrm{isin}\theta}=\frac{\mathrm{1}}{\mathrm{2sin}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)−\mathrm{2isin}\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\theta}{\mathrm{2}}\right)}…

Determine-integers-x-y-z-satisfying-ax-b-by-c-cz-a-a-b-c-are-fixed-integers-

Question Number 4362 by Rasheed Soomro last updated on 13/Jan/16 $$\mathrm{Determine}\:\mathrm{integers}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{satisfying}: \\ $$$$\mathrm{ax}^{\mathrm{b}} +\mathrm{by}^{\mathrm{c}} =\mathrm{cz}^{\mathrm{a}} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{are}\:\mathrm{fixed}\:\mathrm{integers}. \\ $$ Commented by Yozzii last updated on…

Find-that-value-of-2-2-2-continued-power-of-2-using-analytical-continuation-

Question Number 4242 by prakash jain last updated on 05/Jan/16 $$\mathrm{Find}\:\mathrm{that}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}\centerdot\centerdot\centerdot} } \:\:\left(\mathrm{continued}\:\mathrm{power}\:\mathrm{of}\:\mathrm{2}\right) \\ $$$$\mathrm{using}\:\mathrm{analytical}\:\mathrm{continuation}. \\ $$ Commented by RasheedSindhi last updated on…

I-have-no-formal-background-in-number-theory-but-I-m-curious-of-how-to-find-positive-integer-solutions-x-y-z-to-the-equation-x-n-y-n-z-n-for-n-Z-Fermat-s-last-theorem-led-me-to-this-Tell

Question Number 4203 by Yozzii last updated on 01/Jan/16 $${I}\:{have}\:{no}\:{formal}\:{background}\:{in}\: \\ $$$${number}\:{theory},\:{but}\:{I}'{m}\:{curious} \\ $$$${of}\:{how}\:{to}\:{find}\:{positive}\:{integer}\:{solutions}\: \\ $$$$\left({x},{y},{z}\right)\:{to}\:{the}\:{equation}\:{x}^{{n}} +{y}^{{n}} ={z}^{{n}} \:{for}\: \\ $$$${n}\in\mathbb{Z}^{−} .\:{Fermat}'{s}\:{last}\:{theorem}\:{led} \\ $$$${me}\:{to}\:{this}.\:{Tell}\:{me}\:{about}\:{the}\:{cases} \\…