| dbp:quote
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- Diophantine analysis is the study of
equations with rational coefficients
requiring integer solutions. In
Diophantine problems, there are fewer
equations than unknowns. The "extra"
information required to solve the
equations is the condition that the
solutions be integers. Any solution
must satisfy all equations. Some
Diophantine equations have no solution,
some have one or a finite number, and
others have infinitely many solutions.
The monkey and the coconuts reduces to a two-variable linear Diophantine equation of the form
:ax + by = c, or more generally,
:x + y = c/d
where d is the greatest common divisor
of a and b. By Bézout's identity,
the equation is solvable if and only if d divides c. If it does,
the equation has infinitely many periodic
solutions of the form
:x = x0 + tb,
:y = y0 + ta
where is a solution and t is a
parameter than can be any integer. The problem
is not intended to be solved by trial-and-error; there
are deterministic methods for solving
in this case . (en)
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