Extending The Linear Diophantine Problem: a.k.a. 'The Frob prob',Used

Extending The Linear Diophantine Problem: a.k.a. 'The Frob prob',Used

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Brand: LAP Lambert Academic Publishing
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Given integervalued relatively prime coins' a1; a2; :::; ak, the Frobenius number is the largest integer n such that the linear diophantine equation a1m1 + a2m2 + ::: + akmk = n has no solution in nonnegative integers m1;m2; :::;mk. We denote by g(a1; :::; ak) the largest integer value not attainable by this coin system. That is to say that any integer x greater than the Frobenius number g(a1; :::; ak) has a representation x = a1x1 + a2x2 + ::: + akxk by a1; a2; :::; ak for some nonnegative integers x1; x2; :::; xk. We say x is representable by a1; a2; :::; ak. While it is obvious that there are representable positive integers and nonrepresentable positive integers, must there be a largest nonrepresentable integer? Maybe there are indefinitely large nonrepresentable integers for a1; a2; :::; ak with gcd (a1; a2; :::; ak) = 1. This notion of whether or not the Frobenius number is welldefined will be the first bit of mathematics we look at in this paper. Proposition 1.1. The Frobenius number g(a1; :::; ak) is welldefined. Proof. Given a1; a2; :::; ak with gcd (a1; a2; :::; ak) = 1, the extended Euclidean algorithm gives that there exist m1;m2; :::;mk 2 Z such that...

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