How do we find the number of non-negative integer solutions for linear equation of the form:
$$a \cdot x + b \cdot y = c$$
Where $a, b, c$ are constants and $x,y$ are the variables ?
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Joseph Quinsey
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darthy734
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2Generating functions might help - the count is the coefficient of $x^c$ in $\dfrac{1}{(1-x^a)(1-x^b)}$. – Thomas Andrews Jul 22 '13 at 21:03
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Is there a known case where the number of solutions is not either $0$ or $\infty$? If $a,b,c \in \mathbb{Q}$, then any time $b | ka(1-c)$, $k\in\mathbb{N}$, then both $x$ and $y$ are integer solutions and I think there are infinitely many values of $k$ that solve the above. If any of the coefficients are irrational, then there are no integer solutions. – AnonSubmitter85 Jul 22 '13 at 21:15
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@AnonSubmitter85 - Yes, the number of sols is not either 0 or Infinity – darthy734 Jul 22 '13 at 21:17
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1@AnonSubmitter85, note that $x, y \ge 0$, so there are not infinitely many solutions. – George V. Williams Jul 22 '13 at 21:17
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Have you found any answers for specific situations? For instance, if we assume that $a,b,c \in \mathbb{Q}$ and that $y = k \cdot \operatorname{lcm}(b_n,b_d)$, where $b = b_n/b_d$ and $k \in \mathbb{N}$, can you find ranges of $a$ and $c$ such that there are infinitely many or zero solutions? – AnonSubmitter85 Jul 22 '13 at 22:09
2 Answers
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Not a complete answer, but a relatively simple one and approximate one. By Schur's theorem of combinatorics?, the number of solutions is asymptotically ($c \to \infty$):
$$ \frac{c}{ab} $$
Schur's theorem of combinatorics states that the number of solutions of (with $a_i$ relatively prime):
$$ \sum_{i=1}^M a_i x_i = c $$
is:
$$ \frac{c^{M-1}}{(M-1)!\prod a_i} $$
? This name is used by Wilf's Generatingfunctionology, but I cannot seem to find it elsewhere. It appears that Schur has many theorems.
George V. Williams
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@George V. Williams - Thanks, probably this might be the easiest way to get the desired result. Could you pls explain what is 'n' in the above equation ? – darthy734 Jul 22 '13 at 21:28
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@darthy734, my apologies, I wrote $n$ instead of $c$. It's fixed now. – George V. Williams Jul 22 '13 at 22:54
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You don't really need all of Schur for this - it is sort of obvious that $c/ab$ is approximation. – Thomas Andrews Jul 23 '13 at 12:04
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Do you have a source for this? Also, for c=5, M=2, a1=1, a2=2 this gives 2.5... – Ella Sharakanski Apr 26 '15 at 04:41
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@EllaShar, it's asymptotic to the exact number of solutions as $c \to \infty$, not exact. – George V. Williams May 01 '15 at 21:30
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He's not looking for how to solve it, he's looking for how to count non-negative solutions. – Thomas Andrews Jul 22 '13 at 21:05
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Maybe he was looking for small solution sets. Added because it provides a way to count solutions, mon ami. – Torsten Hĕrculĕ Cärlemän Jul 22 '13 at 21:06
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@ThomasAndrews - Can you please throw some more light on the generating functions method. – darthy734 Jul 22 '13 at 21:06
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@AnuragPallaprolu - Going through the link you've shared, will let you know once I'm done. – darthy734 Jul 22 '13 at 21:08
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@AnuragPallaprolu - Thanks for the link, but as Thomas mentioned, is there a way to get the number of non-neg solutions rather than solve for the whole set ? – darthy734 Jul 22 '13 at 21:13
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@darthy734 You could use Generating functions, yes. Please check thislink for an entertaining discussion on Generating functions. This might solve your problem – Torsten Hĕrculĕ Cärlemän Jul 22 '13 at 21:22
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@AnuragPallaprolu - Could you please update the link, it seems to be broken. – darthy734 Jul 22 '13 at 21:25
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@darthy734 Please follow the URL. I think MSE doesn't allow documents to be linked via hyperlink. Copy and paste the link to your browser. Or this. The first link is what I was looking at. – Torsten Hĕrculĕ Cärlemän Jul 22 '13 at 21:26
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