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Question Number 106727 by Algoritm last updated on 06/Aug/20

Answered by Her_Majesty last updated on 06/Aug/20

$$x=1\wedge y=0\Rightarrow min(x+y)=1$$

Commented by Her_Majesty last updated on 06/Aug/20

$$thenextsolutionis$$$$x=9801\wedge y=1820\Rightarrow x+y=11621$$

Commented by mathmax by abdo last updated on 07/Aug/20

$$\mathrm{can}\mathrm{you}\mathrm{show}\mathrm{the}\mathrm{work}...$$

Commented by Her_Majesty last updated on 07/Aug/20

$$x^2-29y^2=1$$$$\iff$$$$y=\sqrt{\frac{x^2-1}{29}}=\sqrt{\frac{(x-1)(x+1)}{29}}$$$$wehavetocancelthe29$$$$1stpossibilityx=\pm 1\Rightarrow y=0$$$$next$$$$x=29n\pm 1\Rightarrow y=\sqrt{29n^2\pm 2n}$$$$andnowwemusttry$$$$n=0$$$$n=338$$$${that}^{\prime }swhatIdid$$

Answered by mathmax by abdo last updated on 06/Aug/20

$$\mathrm{i}\mathrm{have}\mathrm{a}\mathrm{way}\mathrm{but}\mathrm{i}\mathrm{cant}\mathrm{complete}\mathrm{it}\mathrm{that}\mathrm{nead}\mathrm{calculus}$$$$29\mathrm{is}\mathrm{prime}\mathrm{so}\mathrm{we}\mathrm{considere}\mathrm{congruence}\mathrm{modulo}29$$$${\mathrm{x}}^2-{29\mathrm{y}}^2=1\Rightarrow \stackrel{{-}^2}{\mathrm{x}}-0=\stackrel{-}{1}\Rightarrow (\stackrel{-}{\mathrm{x}}-1)(\stackrel{-}{\mathrm{x}}+1)=0\Rightarrow \stackrel{-}{\mathrm{x}}=1\mathrm{or}\stackrel{-}{\mathrm{x}}=-1$$$$\left(\mathrm{Z}/29\mathrm{Z}\right)\mathrm{is}\mathrm{a}\mathrm{corps}\mathrm{case}1\stackrel{-}{\mathrm{x}}=1\Rightarrow \mathrm{x}=29\mathrm{k}+1$$$${\mathrm{x}}^2-{29\mathrm{y}}^2=1\Rightarrow {(29\mathrm{k}+1)}^2-{29\mathrm{y}}^2=1\Rightarrow {29}^2{\mathrm{k}}^2+2.29\mathrm{k}+1-{29\mathrm{y}}^2=1$$$$\Rightarrow {29\mathrm{k}}^2+2\mathrm{k}-{\mathrm{y}}^2=0\Rightarrow {\mathrm{y}}^2={29\mathrm{k}}^2+2\mathrm{k}\Rightarrow \mathrm{y}=\sqrt{{29\mathrm{k}}^2+2\mathrm{k}}$$$$\mathrm{so}\mathrm{we}\mathrm{seark}\mathrm{all}\mathrm{integr}\mathrm{k}/{29\mathrm{k}}^2+2\mathrm{k}={\mathrm{m}}^2$$$$\mathrm{we}\mathrm{verify}\mathrm{with}\mathrm{k}=0,1,2,3,4....\mathrm{be}\mathrm{continued}...$$$$$$

Commented by Algoritm last updated on 07/Aug/20

$$\mathrm{step}\mathrm{by}\mathrm{step}\mathrm{solution}\mathrm{please}\mathrm{sir}$$

Commented by Her_Majesty last updated on 07/Aug/20

$$areyoublind?$$

Commented by Rasheed.Sindhi last updated on 07/Aug/20

$$We{can}^{\prime }texpectanyharsh$$$$commentfromagraceful$$$$personalitysuchasyours!!!$$

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