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$a,b,c$ are natural numbers whose greatest common divisor is $1$.

$a,b,c\in\mathbb{N}^*$, $(a,b,c)=1$

Try to write down the expression using $a,b,c$ of the biggest natural number $M$ that cannot be write down in following forms: $M=xa+yb+zc$, $x,y,z\in\mathbb{N}$


Or say it in this way, $3$ kinds of weights which respectively weigh $a,b,c$, $a,b,c\in\mathbb{N}^*$, $(a,b,c)=1$. They only can be put on one side of the balance. What's the biggest weigh $M\in\mathbb{N}$ that cannot be weighed using these $3$ kinds of weights
What's more, if possible, when there are $a,b,c,\dots,n$ ($n$ kinds of weights in all), what's the largest weigh $M$?
To give some referrings, when there are only two kinds of weights, $M=(a-1)(b-1)$

I think it might be possible to use the linear combinations of primes to solve it but up to now I have no idea.

I'm trying to write a programme that can generate $M$ and put them into coordinate systems and probably I can find the regulation....

Can you help?

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3 Answers 3

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This is known as the Frobenius coin problem. As you state, the solution for one and two numbers is known. For three (or more) only partial results are available.

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In a 2004 paper by Fel titled "Frobenius Problem for Semigroups $S(d_1,d_2,d_3)$ a complete answer to the 3-generator case is given.

We need to define some notation first. $S(d_1,d_2,d_3)$ is the semigroup of all natural numbers that can be obtained using $d_1,d_2,d_3$. $S(d_1,d_2)$ is defined similarly, but only using $d_1, d_2$. Set $c_1$ to be the minimal positive integer such that $c_1d_1\in S(d_2,d_3)$. Similarly, set $c_2$ to be minimal so that $c_2d_2\in S(d_1,d_3)$ and $c_3$ minimal so that $c_3d_3\in S(d_1,d_2)$. Set $w=c_1d_1+c_2d_2+c_3d_3$.

Now, the desired Frobenius number is $\frac{1}{2}\left( w+\sqrt{w^2+4d_1d_2d_3-4(c_3c_2d_3d_2+c_3c_1d_3d_1+c_2c_1d_2d_1)}\right)-d_1-d_2-d_3$.

As a programming matter, however, it's simpler to generate $S(d_1,d_2,d_3[,d_4,\ldots])$ with the knowledge that its frobenius number is no larger than the (known) frobenius number for $S(d_1,d_2)$.

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Just wanted to say that there are two versions of what it means for a set of integers to be all relatively prime, each leading to a slightly different version of this Question:

a) The greatest common divisor of all numbers is 1,

and

b) All pairs of numbers are relatively prime.

The set {6, 10, 15} would be relatively prime under definition a), but not under b). Of course a set satisfying b) would also satisfy a).

Obviously a) is the definition used in this Question. But using definition b) might also be interesting.

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