Diophantine Equation

• February 8th 2010, 05:21 PM
CoraGB
Diophantine Equation
I was looking for some help solving the equation
15 625x + 8404 = 1024y

thank you
• February 8th 2010, 07:01 PM
tonio
Quote:

Originally Posted by CoraGB
I was looking for some help solving the equation
15 625x + 8404 = 1024y

thank you

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

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

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

Tonio