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Let $J$ be a set of nonnegative integers whose greatest common divisor is $d$. And suppose that $J$ is closed under addition, then J contains all but a finite number of integers in the set $ \{ 0,d,2d,3d,... \}$. I could hardly believe in this, I hope one of you could give me a proof on the statement mentioned above. Thanks in advance!

davidlowryduda
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luckystarufo
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  • What are your thoughts on this problem? I'm having a hard time seeing the relation to stochastic processes. – Tyler Oct 22 '13 at 16:48
  • This is just another way to state the finiteness of the Frobenius number: https://en.wikipedia.org/wiki/Coin_problem The same upper bound as in user940's answer can be found here: https://math.stackexchange.com/questions/1134666 – Bart Michels Oct 24 '17 at 18:25

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Choose $j_1,\dots, j_r\in J$ with $\gcd(j_1,\dots, j_r)=d$. The set $J$ contains all positive linear combinations $p_1j_1+\cdots+p_rj_r$ where the $p_i$'s are non-negative integers.

Claim: $J$ contains all sufficiently large multiples of $d$.

Proof. By Euclid's algorithm there are integers $c_j$ such that $$ c_1j_1+\cdots+c_rj_r =d.$$ Put $ s = j_1 + j_2 + \cdots + j_r$. Every integer $n$ has a representation $$ n = xs + y $$ where $x$ and $y$ are integers and $0 \leq y < s$. Then $$ nd = \sum_{k=1}^r ( dx + c_ky)j_k$$ and all the coefficients are positive if $dx > \max_k |c_k| s$.