THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS FOR SUBSETS OF ARITHMETICSEQUENCES

Let A(k) = {a(1),...,a(k)} subset of N with gcd(a1,..., a(k)) = 1. We shall say that a natural number n has a representation by a(,)..., a(k ) if n = (i=1)Sigma(k) a(i)x(i), x(i) is an element of N-0. Let g = g( A(k)) be the largest integer with no such representation. We then stud y the set A(k) = {a, ha + d, ha + 2d,..., ha + (k - 1)d} (h, d > 0, gc d(a, d) = 1). If l(k) denotes the greatest number of elements which ca n be omitted without altering g(Ak), we show that 1-=4/root k less tha n or equal to l(k)/l less than or equal to 1 3/k provided a > k, or a = k with d > 2h root k. The lower bound can be improved to 1 - 4/k if we choose a > (k-4)k+3. Moreover, we determine sets E-k subset of A(k) such that g(Ak\Ek) = g(Ak).