I was looking for some help solving the equation
15 625x + 8404 = 1024y
thank you
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