Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 112575 by Aina Samuel Temidayo last updated on 08/Sep/20

Answered by Rasheed.Sindhi last updated on 09/Sep/20

x^2 −3y^2 =1376 ∵4∣1376 ∴ 4∣x^2 −3y^2 case1:x,y∈O x=2m+1,y=2n+1 x^2 −3y^2 =(2m+1)^2 −3(2n^2 +1)^2 =4m^2 +4m+1−12n^2 −12n−3 =4(m^2 +m−3n^2 −3n)−2 ∴ 4 ∤ x^2 −3y^2 case2: x∈O ∧ y∈E or x∈E ∧ y∈O Similar logic proves that 4 ∤ x^2 −3y^2 case1:x,y∈E Similar logic proves that 4 ∣ x^2 −3y^2 Let x=2m,y=2n x^2 −3y^2 =1376 ⇒(2m)^2 −3(2n)^2 =1376 ⇒4m^2 −12n^2 =1376 ⇒m^2 −3n^2 =344 Again 4∣344⇒4∣m^2 −3n^2 By similar process Let m=2u,n=2v u^2 −3v^2 =86 u=2p+1,v=2q+1 u^2 −3v^2 =86 u= (2p+1)^2 −3(2q+1)^2 =86 4p^2 +4p−12q^2 −12q−2=86 4p^2 +4p−12q^2 −12q=88 p^2 +p−3(q^2 +q)=22 p(p+1)−3q(q+1)=22 p(p+1)∈E ∧ q(q+1)∈E p(p+1)=2a,q(q+1)=2b◂ ⇒2a−3(2b)=22 a−3b=11 a=3b+11 (a,b)=(3b+11,b) Continue

$${x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} =\mathrm{1376} \\ $$$$\because\mathrm{4}\mid\mathrm{1376}\:\:\:\:\:\therefore\:\mathrm{4}\mid{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \\ $$$${case}\mathrm{1}:{x},{y}\in\mathbb{O} \\ $$$${x}=\mathrm{2}{m}+\mathrm{1},{y}=\mathrm{2}{n}+\mathrm{1} \\ $$$${x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} =\left(\mathrm{2}{m}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{3}\left(\mathrm{2}{n}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:=\mathrm{4}{m}^{\mathrm{2}} +\mathrm{4}{m}+\mathrm{1}−\mathrm{12}{n}^{\mathrm{2}} −\mathrm{12}{n}−\mathrm{3} \\ $$$$=\mathrm{4}\left({m}^{\mathrm{2}} +{m}−\mathrm{3}{n}^{\mathrm{2}} −\mathrm{3}{n}\right)−\mathrm{2} \\ $$$$\therefore\:\mathrm{4}\:\nmid\:{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \\ $$$${case}\mathrm{2}:\:{x}\in\mathbb{O}\:\wedge\:{y}\in\mathbb{E}\:{or}\:{x}\in\mathbb{E}\:\wedge\:{y}\in\mathbb{O} \\ $$$${Similar}\:{logic}\:{proves}\:{that} \\ $$$$\:\:\:\:\:\:\mathrm{4}\:\nmid\:{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \\ $$$${case}\mathrm{1}:{x},{y}\in\mathbb{E} \\ $$$${Similar}\:{logic}\:{proves}\:{that} \\ $$$$\:\:\:\:\:\:\mathrm{4}\:\mid\:{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \\ $$$${Let}\:{x}=\mathrm{2}{m},{y}=\mathrm{2}{n} \\ $$$${x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} =\mathrm{1376} \\ $$$$\Rightarrow\left(\mathrm{2}{m}\right)^{\mathrm{2}} −\mathrm{3}\left(\mathrm{2}{n}\right)^{\mathrm{2}} =\mathrm{1376} \\ $$$$\Rightarrow\mathrm{4}{m}^{\mathrm{2}} −\mathrm{12}{n}^{\mathrm{2}} =\mathrm{1376} \\ $$$$\Rightarrow{m}^{\mathrm{2}} −\mathrm{3}{n}^{\mathrm{2}} =\mathrm{344} \\ $$$$\:\:{Again}\:\mathrm{4}\mid\mathrm{344}\Rightarrow\mathrm{4}\mid{m}^{\mathrm{2}} −\mathrm{3}{n}^{\mathrm{2}} \\ $$$$\:{By}\:{similar}\:{process} \\ $$$$\:\:\:\:\:{Let}\:{m}=\mathrm{2}{u},{n}=\mathrm{2}{v} \\ $$$$\:\:\:\:\:\:{u}^{\mathrm{2}} −\mathrm{3}{v}^{\mathrm{2}} =\mathrm{86} \\ $$$${u}=\mathrm{2}{p}+\mathrm{1},{v}=\mathrm{2}{q}+\mathrm{1} \\ $$$$\:{u}^{\mathrm{2}} −\mathrm{3}{v}^{\mathrm{2}} =\mathrm{86} \\ $$$${u}= \\ $$$$\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{3}\left(\mathrm{2}{q}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{86} \\ $$$$\mathrm{4}{p}^{\mathrm{2}} +\mathrm{4}{p}−\mathrm{12}{q}^{\mathrm{2}} −\mathrm{12}{q}−\mathrm{2}=\mathrm{86} \\ $$$$\mathrm{4}{p}^{\mathrm{2}} +\mathrm{4}{p}−\mathrm{12}{q}^{\mathrm{2}} −\mathrm{12}{q}=\mathrm{88} \\ $$$${p}^{\mathrm{2}} +{p}−\mathrm{3}\left({q}^{\mathrm{2}} +{q}\right)=\mathrm{22} \\ $$$${p}\left({p}+\mathrm{1}\right)−\mathrm{3}{q}\left({q}+\mathrm{1}\right)=\mathrm{22} \\ $$$${p}\left({p}+\mathrm{1}\right)\in\mathbb{E}\:\wedge\:{q}\left({q}+\mathrm{1}\right)\in\mathbb{E} \\ $$$${p}\left({p}+\mathrm{1}\right)=\mathrm{2}{a},{q}\left({q}+\mathrm{1}\right)=\mathrm{2}{b}\blacktriangleleft \\ $$$$\Rightarrow\mathrm{2}{a}−\mathrm{3}\left(\mathrm{2}{b}\right)=\mathrm{22} \\ $$$$\:\:\:\:{a}−\mathrm{3}{b}=\mathrm{11} \\ $$$$\:\:\:\:{a}=\mathrm{3}{b}+\mathrm{11} \\ $$$$\left({a},{b}\right)=\left(\mathrm{3}{b}+\mathrm{11},{b}\right) \\ $$$${Continue} \\ $$$$ \\ $$

Commented by Aina Samuel Temidayo last updated on 09/Sep/20

There are no solutions.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{no}\:\mathrm{solutions}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com