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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