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Sunday, February 23, 2014

Linear diophantine equations: simplification of sums for reachability (enabled javascipt for latex is needed)

My first meaningful post shall be a mathy one. I wanted to understand the idea behind this, so I've decided to make a post explaining it :)
Assume we have a problem where we have to iterate over all \(x = \sum\limits_{i=1}^n c_{i} \cdot a_{i} \), where the \(c_{i}\) are freely selectable. In such problems a very easy simplification can be made: each such \(x\) can be represented in the form \(x = t \cdot q\), where \(t = gcd(a_{1},...,a_{n})\) and \(q \in \mathbb Z \).

To understand the above statement, we only need to think of the extended euclidean algorithm. Inside of it we find the integers \(x\) and \(y\) so that \(x \cdot a_{1} + y \cdot a_{2} = gcd(a_{1},a_{2}) \) holds. Additionally, we can easily imply that each number \(v\) written in the form \(v = c_{1} \cdot a_{1} + c_{2} \cdot a_{2} \) must be a multiple of \(g:=gcd(a_{1},a_{2})\) (as g divides \(a_{1}\)and \(a_{2}\) it must divide v for the equation to hold (!)).
We show now that the same holds for 3 variables. Let \(x = a_{1} \cdot c_{1} + a_{2} \cdot c_{2} + a_{3} \cdot c_{3} \) and \(g:=gcd(a_{1},a_{2},a_{3})\). We have to prove that \(x\) is a multiple of \(g\). Consider the first two variables and write: \(a_{1} \cdot c_{1} + a_{2} \cdot c_{2} = t_{1} \cdot gcd(a_{1},a_{2})\). This gives us \(x = a_{1} \cdot c_{1} + a_{2} \cdot c_{2} + a_{3} \cdot c_{3} = t_{1} \cdot gcd(a_{1},a_{2}) + a_{3} \cdot c_{3} \). Simply seen here: there are again two fixed numbers and two free! Smells like the extended euclidean again :)
We write: \(G := gcd( gcd(a_{1},a_{2}) , a_{3} ) = gcd(a_{1},a_{2},a_{3}) = g \) and get: \(x = t_{1} \cdot gcd(a_{1},a_{2}) + a_{3} \cdot c_{3} = t_{2} \cdot g \) - with this we're done! Inductively the statement follows for 4 and more variables.

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