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 110644 by Aina Samuel Temidayo last updated on 29/Aug/20

The Diophantine equation  x^2 +y^2 +1 =N(xy+1) has  infinitely many integer  solutions if N equals?  Any help please?

$$\mathrm{The}\:\mathrm{Diophantine}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}\:=\mathrm{N}\left(\mathrm{xy}+\mathrm{1}\right)\:\mathrm{has} \\ $$$$\mathrm{infinitely}\:\mathrm{many}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{if}\:\mathrm{N}\:\mathrm{equals}? \\ $$$$\mathrm{Any}\:\mathrm{help}\:\mathrm{please}? \\ $$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

No one has been able to solve this?

$$\mathrm{No}\:\mathrm{one}\:\mathrm{has}\:\mathrm{been}\:\mathrm{able}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{this}? \\ $$

Answered by floor(10²Eta[1]) last updated on 30/Aug/20

if N=2  x^2 +y^2 +1=2xy+2  (x−y)^2 =1⇒x−y=±1

$$\mathrm{if}\:\mathrm{N}=\mathrm{2} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}=\mathrm{2xy}+\mathrm{2} \\ $$$$\left(\mathrm{x}−\mathrm{y}\right)^{\mathrm{2}} =\mathrm{1}\Rightarrow\mathrm{x}−\mathrm{y}=\pm\mathrm{1} \\ $$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

Yes but how did you arrive at N=2?

$$\mathrm{Yes}\:\mathrm{but}\:\mathrm{how}\:\mathrm{did}\:\mathrm{you}\:\mathrm{arrive}\:\mathrm{at}\:\mathrm{N}=\mathrm{2}? \\ $$

Commented by floor(10²Eta[1]) last updated on 30/Aug/20

i mean- can′t you see that x^2 +y^2 −2xy=(x−y)^2 ??

$$\mathrm{i}\:\mathrm{mean}-\:\mathrm{can}'\mathrm{t}\:\mathrm{you}\:\mathrm{see}\:\mathrm{that}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{2xy}=\left(\mathrm{x}−\mathrm{y}\right)^{\mathrm{2}} ?? \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com