{"name":"congruence","title":"Solves mx+by=1 for x and y","pagetitle":"Solves mx+by=1 for x and y — congruence","aliases":"congruence","author":"Robin K. S. Hankin","keywords":"math","description":"

Solves the Diophantine equation \\(mx+by=1\\) for \\(x\\)\n and \\(y\\). The function is named for equation 57 in Hardy and Wright.

","usage":"congruence(a, l = 1)","arguments":[{"name":"a","description":"

Two element vector with a=c(m,n)

"},{"name":"l","description":"

Right hand side with default 1

"}],"has_args":true,"examples":"# NOT RUN {\nM <- congruence(c(4,9))\ndet(M)\n\no <- c(1,1i)\ng2.fun(o) - g2.fun(o,maxiter=840) #should be zero\n\n# }\n","sections":[],"value":"

In the usual case of \\((m,n)=1\\), returns a square matrix\nwhose rows are a and c(x,y). This matrix is a unimodular\ntransformation that takes a pair of basic periods to another pair of\nbasic periods.

If \\((m,n)\\neq 1\\) then more than one solution is\navailable (for example congruence(c(4,6),2)). In this case, extra rows\nare added and the matrix is no longer square.

","references":"

G. H. Hardy and E. M. Wright 1985. An introduction to the\n theory of numbers, Oxford University Press (fifth edition)

","note":"

This function does not generate all unimodular matrices with a\n given first row (here, it will be assumed that the function returns a\n square matrix).

For a start, this function only returns matrices all of whose\n elements are positive, and if a is unimodular, then after\n diag(a) <- -diag(a), both a and -a are\n unimodular (so if a was originally generated by\n congruence(), neither of the derived matrices could be).

Now if the first row is c(1,23), for example, then the second\n row need only be of the form c(n,1) where n is any\n integer. There are thus an infinite number of unimodular matrices\n whose first row is c(1,23). While this is (somewhat)\n pathological, consider matrices with a first row of, say,\n c(2,5). Then the second row could be c(1,3), or\n c(3,8) or c(5,13). Function congruence() will\n return only the first of these.

To systematically generate all unimodular matrices, use\n unimodular(), which uses Farey sequences.

","seealso":"

unimodular

","package":{"package":"elliptic","version":"1.4-0"}}

Description

Usage

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Examples