$(m,n)=1$,$N=am+bn$,where $a ,b$ are non-negative integers. what's the max integer $N$,that cannot expressed as $N=am+bn$?

Asked Jan 30 '11 at 05:37

Active Dec 06 '11 at 19:00

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When do the multiples of two primes span all large enough natural numbers?

If $m,n$ are positive integers and $(m,n)=1$.

What's the largest integer $N$ that cannot be expressed as $N=am+bn$, where $a$ and $b$ are nonnegative integers?

The answer is $N=mn-m-n$. How to prove it?

edited Apr 13 '17 at 12:19

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asked Jan 30 '11 at 05:37

This is basically a duplicate of http://math.stackexchange.com/questions/8186/when-do-the-multiples-of-two-primes-span-all-large-enough-natural-numbers – Mike Spivey Jan 30 '11 at 05:43
Another question asked what $N$ is, but Qiaochu Yuan's answer there gives a proof as well: http://math.stackexchange.com/questions/8186/when-do-the-multiples-of-two-primes-span-all-large-enough-natural-numbers/8187#8187 – Jonas Meyer Jan 30 '11 at 05:43
This is the $n=2$ case of the Frobenius problem. http://en.wikipedia.org/wiki/Frobenius_problem" (also known as the "coin problem"). The result you quote is due to Sylvester. – Arturo Magidin Jan 30 '11 at 05:44
so shall we close it. – Jan 30 '11 at 05:48

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