Frobenius Equation in Two Variables

The Frobenius equation in two variables is a Diophantine equation , where and . The Frobenius number of the coefficients and , where and are relatively prime, is the largest for which the equation has no non-negative solutions. Sylvester (1884) showed that .
The equation has the intercept form and only two non-negative solutions and (brown points). The difference between the solutions (as vectors) is .
The Diophantine equation , where and are relatively prime, has at least one solution, and the difference between two consecutive solutions is . If , , and , the equation has, because of this difference, at least one non-negative solution.
The equation can be written in the form and has solutions and (the magenta points). It has no non-negative solution. Any equation , has exactly one non-negative solution (the green point). It is inside the parallelogram determined by brown and magenta points.


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