# Extensions of Chicken Nugget Theorem

Given that $$a,b,c\$$ are pairwise relatively prime, what is the largest number not expressible as a linear combination of $$a,b,c\$$?

What if $$a$$ and $$b$$ have common factor $$m$$?

What if $$a$$ and $$b$$ have common factor $$m$$, and $$b$$ and $$c$$ have common factor $$n$$?

• I don’t know how to do the notation correctly. Can anyone help me with that? – Lieutenant Zipp Apr 25 '19 at 21:22
• For starters, put dollar signs at each end of each attempted MathJax expression. – J.G. Apr 25 '19 at 21:23
• Out of curiousity...why do they call it the Chicken Nugget Theorem? – Mike Apr 25 '19 at 21:27
• – bounceback Apr 25 '19 at 21:29
• There was a Numberphile video on it a while back. I think it originally comes from people trying to order McDonald’s nuggets which came in boxes of 9 and 20. Or something. – Lieutenant Zipp Apr 25 '19 at 21:30

This is more of an extended comment than an answer to your specific questions.

This is called the Frobenius number of the semigroup generated by $$a,b$$ and $$c$$. The case for two generators, say $$a$$ and $$b$$ with $$a$$ and $$b$$ rel. prime, has a closed formula given by $$g(a,b)=ab-a-b.$$

However, in general for 3 generators it is known that there is no polynomial formula expressing the Frobenius number in terms of $$a,b$$ and $$c$$. The reference is F. Curtis (1990). "On formulas for the Frobenius number of a numerical semigroup". Mathematica Scandinavica. 67 (2): 190–192. You can find the paper here: https://www.mscand.dk/article/view/12330/10346

Your not all relatively prime cases, are restricted cases of the 2 number cases: $$a=md\land b=me \implies m(dx+ey)+cz=f$$