this is called Frobenius's Problem, of course (I was expecting many people to say that, but I didn't see that mentioned here) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Dec 21, 2020 at 3:09 AM christopher landauer <topcycal@gmail.com> wrote:
hihi, all -
ah, excellent, thank you, fred - i wrote a computer program (of course) this evening that suggested the sylvester result, but did not come close to proving it, and i assume that that result sort of implies the general result to answer dan's question for more than two elements (i imagine that a suitable induction on r would work):
at most finitely many positive integers n divisible by c = gcd({N_j|1<=j<=r}) are not representable in the form n = sum(1<=j<=r) (c_j * N_j), c_j non-negative integers
more later,
chris
On 12/20/20 23:43, Fred Lunnon wrote:
See Conway's money game --- https://en.wikipedia.org/wiki/Sylver_coinage
WFL
On 12/21/20, christopher landauer <topcycal@gmail.com> wrote:
hihi -
it is my vague recollection that for positive integers a, b with gcd(a, b) = 1,
then for all n >= a * b,
there is a representation n = x * a + y * b with x, y >= 0
(this is trivial for a=1 or b=1, so we can assume a, b >= 2)
that would mean density 1
(but this result is even stronger than the original density assertion:
it says that all but finitely many positive integers have such a representation;
i think i saw a reasonable proof of that once long ago)
it is my impression that essentially the same kind of thing
could/should/would be true for more than two elements:
if we take c = gcd{N_j},
then for any large enough n divisible by c,
there is a representation n = sum(x_j * N_j) with all x_j >= 0,
so the density is 1/c
(only multiples of c can occur, of course,
and the stronger assertion is that all but finitely many of them have such a representation)
more later,
chris
On 12/18/20 23:25, Dan Asimov wrote:
Let N_1, ..., N_r be a set of positive integers ≥ 2 whose prime factorizations are known.
Let X = {∑ c_j N_j} be the set of all linear combinations of the N_j with nonnegative integer coefficients c_j.
Is it easy to determine what the (asymptotic) density of X is in Z+ ???
(An N_j may have repeated prime factors, and several N_j's may have common factors.)
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