I see in this document the following method to solve the Diophantine equation $1234x+2341y=1$:
It looks pretty useful and interesting, but I don't know what the cited work MNZ p.218 is. Can anyone tell me how this method works?
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Ma Joad
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https://stattrek.com/matrix-algebra/elementary-operations.aspx – thesmallprint Dec 01 '19 at 14:41
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The same method is described here in my answer in How to use the Extended Euclidean Algorithm manually?. You can find $\approx 150$ worked examples in the Linked Questions there. For generalizations see the keywords listed there, esp. Hermite / Smith normal form. See also the slicker fractional form – Bill Dubuque Dec 01 '19 at 16:26
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MNZ refers to Montgomery, Niven, Zuckerman. Your method is presented in Chapter 5, Section 1, pages 212-218 in the Fifth Edition. Your example begins with two coprime numbers, so the final diagram finishes the problem. The problem shown at the top of page 218 deals with a gcd of 3, a little extra work needed.
Instead of row operations, they are using column operations, building one step at a time. Begin with row vector $r = (1234, 2341)$ and construct square matrix $M$ such that $rM = (0,1).$ Here we have constructed $\det M = \pm 1.$ It follows that the dot product of $r$ with the right hand column of $M$ is $1.$ Furthermore, the dot product of $r$ with the left column of $M$ is zero, so we may freely add any multiple (they use $t$) of the left column of $M$ and still have a solution.
Will Jagy
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