 Reordering,

This Diophantine equation has a solution which follows directly from the Euclidean algorithm; in fact, it has infinitely many periodic solutions positive and negative.If (x0, y0) is a solution of 1024x–15625y=1, 
then N0=x08404, F0=y08404 is a solution of (2), which means any solution must have the form

where formula_16 is an arbitrary parameter that can have any integral value.One can take both sides of (1) above modulo 1024, so
Another way of thinking about it is that in order for formula_3 to be an integer, the RHS of the equation must be an integral multiple of 1024; that property will be unaltered by factoring out as many multiples of 1024 as possible from the RHS.