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Math Help - Diophantine Equation

  1. #1
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    Diophantine Equation

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

    thank you
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  2. #2
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    Quote Originally Posted by CoraGB View Post
    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
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