# Frobenius Equation in Two Variables

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 .

ax+by=c

a>0

b>0

a

b

a

b

c

ax+by=c

c=ab-a-b

The equation has the intercept form and only two non-negative solutions and (brown points). The difference between the solutions (as vectors) is .

ax+by=ab

x/b+y/a=1

(b,0)

(0,a)

(b,-a)

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.

ax+by=c

a

b

±(b,-a)

a>0

b>0

c≥ab

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.

ax+by=ab-a-b

(x+1)/b+(y+1)/b=1

(b-1,-1)

(-1,a-1)

ax+by=ab-d

d=1,...,a+b-1