I was looking for some help solving the equation

15 625x + 8404 = 1024y

thank you

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- Feb 8th 2010, 05:21 PMCoraGBDiophantine Equation
I was looking for some help solving the equation

15 625x + 8404 = 1024y

thank you - Feb 8th 2010, 07:01 PMtonio

You have $\displaystyle 5^6x+2^2\cdot 11\cdot 191=2^{10}y$ $\displaystyle \Longrightarrow 5^6x-2^{10}y=-2^2\cdot 11\cdot 191$ .

Since $\displaystyle (5^6,2^{10})=1$ , the above equation has solution in the integers, and as $\displaystyle 5^6\cdot 313+2^{10}\cdot (-4776)=1$, we get that $\displaystyle 5^6(313\cdot T)+2^{10}(-4776\cdot T)=T$ , for any

integer $\displaystyle T$ and the solution is then $\displaystyle x=313\,T\,,\,\,y=-4776\,T$ . Well, now just input $\displaystyle T=8404=2^2\cdot 11\cdot 191$ and we're done.

Tonio