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Topic: Linear Combination of Non-negative integers.
Replies: 4   Last Post: Aug 26, 2000 11:24 AM

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Robert Hill

Posts: 529
Registered: 12/8/04
Re: Linear Combination of Non-negative integers.
Posted: Aug 22, 2000 4:46 PM
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In article <39A24296.52BC@club-internet.fr>, Pierre Bornsztein <pbornszt@club-internet.fr> writes:

> according to Guy's "unsolved problems in number theory" Springer p.113
> (C7), this result is due to Sylvester (1884), who also proved that the
> number of non-representable numbers is (a-1)(b-1)/2.
> The general problem with n > 1 numbers is known as the coin exchange
> problem of Frobenius.
> The case n = 3 has been solved by Selmer and Beyer, then simplified by
> Rödseth and later by Greenberg. But there is no formula as simple as
> above.
> For n > 3, only bounds are known.


This is the basis of Conway's game of "Sylver Coinage", named in honour
of Sylvester (see Berlekamp, Conway and Guy, "Winning Ways", vol. 2).

The basic results for n=2 numbers are so easy (for example, I was able
to independently rediscover and prove them, and I'm pretty dim) that I
wonder if somebody knew them before Sylvester (whose contribution was
not limited to the case n=2). Maybe an earlier publication has been
overlooked, or somebody knew the stuff but didn't trouble to publish it.

--
Robert Hill





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