6533b82dfe1ef96bd1291f82
RESEARCH PRODUCT
On a Linear Diophantine Problem of Frobenius: Extending the Basis
Stefan Matthias Rittersubject
CombinatoricsDiscrete mathematicsAlgebra and Number TheoryCardinalityIntegerDiophantine equationArithmetic progressionValue (computer science)Basis (universal algebra)Element (category theory)Mathematicsdescription
LetXk={a1, a2, …, ak},k>1, be a subset of N such that gcd(Xk)=1. We shall say that a natural numbernisdependent(onXk) if there are nonnegative integersxisuch thatnhas a representationn=∑ki=1 xiai, elseindependent. The Frobenius numberg(Xk) ofXkis the greatest integer withnosuch representation. Selmer has raised the problem of extendingXkwithout changing the value ofg. He showed that under certain conditions it is possible to add an elementc=a+kdto the arithmetic sequencea,a+d,a+2d, …, a+(k−1) d, gcd(a, d)=1, without alteringg. In this paper, we give the setCof all independent numberscsatisfyingg(A, c)=g(A), whereAcontains the elements of the arithmetic sequence. Moreover, ifa>kthen we give as an application, a setBof maximal cardinality such thatg(A, B)=g(A) and each element ofA∪Bis independent of the other ones.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 1998-04-01 | Journal of Number Theory |