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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::GcdRationals \-\- rationals by triangular GCD .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::GcdRationals; \& my $path = Math::PlanePath::GcdRationals\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path enumerates X/Y rationals using a method by Lance Fortnow taking a greatest common divisor out of a triangular position. .IX Xref "Fortnow, Lance" .Sp .RS 4 .RE .PP The attraction of this approach is that it's both efficient to calculate and visits blocks of X/Y rationals using a modest range of N values, roughly a square N=2*max(num,den)^2 in the default rows style. .PP .Vb 10 \& 13 | 79 80 81 82 83 84 85 86 87 88 89 90 \& 12 | 67 71 73 77 278 \& 11 | 56 57 58 59 60 61 62 63 64 65 233 235 \& 10 | 46 48 52 54 192 196 \& 9 | 37 38 40 41 43 44 155 157 161 \& 8 | 29 31 33 35 122 126 130 \& 7 | 22 23 24 25 26 27 93 95 97 99 101 103 \& 6 | 16 20 68 76 156 \& 5 | 11 12 13 14 47 49 51 53 108 111 114 \& 4 | 7 9 30 34 69 75 124 \& 3 | 4 5 17 19 39 42 70 74 110 \& 2 | 2 8 18 32 50 72 98 \& 1 | 1 3 6 10 15 21 28 36 45 55 66 78 91 \& Y=0 | \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 .Ve .PP The mapping from N to rational is .PP .Vb 3 \& N = i + j*(j\-1)/2 for upper triangle 1 <= i <= j \& gcd = GCD(i,j) \& rational = i/j + gcd\-1 .Ve .PP which means X=numerator Y=denominator are .PP .Vb 2 \& X = (i + j*(gcd\-1))/gcd = j + (i\-j)/gcd \& Y = j/gcd .Ve .PP The i,j position is a numbering of points above the X=Y diagonal by rows in the style of Math::PlanePath::PyramidRows with step=1, but starting from i=1,j=1. .PP .Vb 6 \& j=4 | 7 8 9 10 \& j=3 | 4 5 6 \& j=2 | 2 3 \& j=1 | 1 \& +\-\-\-\-\-\-\-\-\-\-\-\-\- \& i=1 2 3 4 .Ve .PP If \s-1GCD\s0(i,j)=1 then X/Y is simply X=i,Y=j unchanged. This means fractions X/Y\ <\ 1 are numbered by rows with increasing numerator, but skipping positions where i,j have a common factor. .PP The skipped positions where i,j have a common factor become rationals X/Y>1, ie. below the X=Y diagonal. The integer part is \s-1GCD\s0(i,j)\-1 so rational\ =\ gcd\-1\ +\ i/j. For example .PP .Vb 4 \& N=51 is at i=6,j=10 by rows \& common factor gcd(6,10)=2 \& so rational R = 2\-1 + 6/10 = 1+3/5 = 8/5 \& ie. X=8,Y=5 .Ve .PP If j is prime then gcd(i,j)=1 and so X=i,Y=j. This means that in rows with prime Y are numbered by consecutive N across to the X=Y diagonal. For example in row Y=7 above N=22 to N=27. .SS "Triangular Numbers" .IX Subsection "Triangular Numbers" N=1,3,6,10,etc along the bottom Y=1 row is the triangular numbers N=k*(k\-1)/2. Such an N is at i=k,j=k and has gcd(i,j)=k which divides out to Y=1. .IX Xref "Triangular numbers" .PP .Vb 1 \& N=k*(k\-1)/2 i=k,j=k \& \& Y = j/gcd \& = 1 on the bottom row \& \& X = (i + j*(gcd\-1)) / gcd \& = (k + k*(k\-1)) / k \& = k\-1 successive points on that bottom row .Ve .PP N=1,2,4,7,11,etc in the column at X=1 immediately follows each of those bottom row triangulars, ie. N+1. .PP .Vb 1 \& N in X=1 column = Y*(Y\-1)/2 + 1 .Ve .SS "Primes" .IX Subsection "Primes" If N is prime then it's above the sloping line X=2*Y. If N is composite then it might be above or below, but the primes are always above. Here's the table with dots \*(L"...\*(R" marking the X=2*Y line. .PP .Vb 10 \& primes and composites above \& | \& 6 | 16 20 68 \& | .... X=2*Y \& 5 | 11 12 13 14 47 49 51 53 .... \& | .... \& 4 | 7 9 30 34 .... 69 \& | .... \& 3 | 4 5 17 19 .... 39 42 70 only \& | .... composite \& 2 | 2 8 .... 18 32 50 below \& | .... \& 1 | 1 ..3. 6 10 15 21 28 36 45 55 \& | .... \& Y=0 | .... \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 10 .Ve .PP Values below X=2*Y such as 39 and 42 are always composite. Values above such as 19 and 30 are either prime or composite. Only X=2,Y=1 is exactly on the line, which is prime N=3 as it happens. The rest of the line X=2*k,Y=k is not visited since common factor k would mean X/Y is not a rational in least terms. .PP This pattern of primes and composites occurs because N is a multiple of gcd(i,j) when that gcd is odd, or a multiple of gcd/2 when that gcd is even. .PP .Vb 2 \& N = i + j*(j\-1)/2 \& gcd = gcd(i,j) \& \& N = gcd * (i/gcd + j/gcd * (j\-1)/2) when gcd odd \& gcd/2 * (2i/gcd + j/gcd * (j\-1)) when gcd even .Ve .PP If gcd odd then either j/gcd or j\-1 is even, to take the \*(L"/2\*(R" divisor. If gcd even then only gcd/2 can come out as a factor since taking out the full gcd might leave both j/gcd and j\-1 odd and so the \*(L"/2\*(R" not an integer. That happens for example to N=70 .PP .Vb 4 \& N = 70 \& i = 4, j = 12 for 4 + 12*11/2 = 70 = N \& gcd(i,j) = 4 \& but N is not a multiple of 4, only of 4/2=2 .Ve .PP Of course knowing gcd or gcd/2 is a factor of N is only useful when that factor is 2 or more, so .PP .Vb 2 \& odd gcd >= 2 means gcd >= 3 \& even gcd with gcd/2 >= 2 means gcd >= 4 \& \& so N composite when gcd(i,j) >= 3 .Ve .PP If gcd<3 then the \*(L"factor\*(R" coming out is only 1 and says nothing about whether N is prime or composite. There are both prime and composite N with gcd<3, as can be seen among the values above the X=2*Y line in the table above. .SS "Rows Reverse" .IX Subsection "Rows Reverse" Option \f(CW\*(C`pairs_order => "rows_reverse"\*(C'\fR reverses the order of points within the rows of i,j pairs, .PP .Vb 6 \& j=4 | 10 9 8 7 \& j=3 | 6 5 4 \& j=2 | 3 2 \& j=1 | 1 \& +\-\-\-\-\-\-\-\-\-\-\-\- \& i=1 2 3 4 .Ve .PP The X,Y numbering becomes .PP .Vb 1 \& pairs_order => "rows_reverse" \& \& 11 | 66 65 64 63 62 61 60 59 58 57 \& 10 | 55 53 49 47 209 \& 9 | 45 44 42 41 39 38 170 168 \& 8 | 36 34 32 30 135 131 \& 7 | 28 27 26 25 24 23 104 102 100 98 \& 6 | 21 17 77 69 \& 5 | 15 14 13 12 54 52 50 48 118 \& 4 | 10 8 35 31 76 70 \& 3 | 6 5 20 18 43 40 75 71 \& 2 | 3 9 19 33 51 73 \& 1 | 1 2 4 7 11 16 22 29 37 46 56 \& Y=0 | \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 10 11 .Ve .PP The triangular numbers, per \*(L"Triangular Numbers\*(R" above, are now in the X=1 column, ie. at the left rather than at the Y=1 bottom row. That bottom row is now the next after each triangular, ie. T(X)+1. .SS "Diagonals" .IX Subsection "Diagonals" Option \f(CW\*(C`pairs_order => "diagonals_down"\*(C'\fR takes the i,j pairs by diagonals down from the Y axis. \f(CW\*(C`pairs_order => "diagonals_up"\*(C'\fR likewise but upwards from the X=Y centre up to the Y axis. (These numberings are in the style of Math::PlanePath::DiagonalsOctant.) .PP .Vb 1 \& diagonals_down diagonals_up \& \& j=7 | 13 j=7 | 16 \& j=6 | 10 14 j=6 | 12 15 \& j=5 | 7 11 15 j=5 | 9 11 14 \& j=4 | 5 8 12 16 j=4 | 6 8 10 13 \& j=3 | 3 6 9 j=3 | 4 5 7 \& j=2 | 2 4 j=2 | 2 3 \& j=1 | 1 j=1 | 1 \& +\-\-\-\-\-\-\-\-\-\-\-\- +\-\-\-\-\-\-\-\-\-\-\-\- \& i=1 2 3 4 i=1 2 3 4 .Ve .PP The resulting path becomes .PP .Vb 1 \& pairs_order => "diagonals_down" \& \& 9 | 21 27 40 47 63 72 \& 8 | 17 28 41 56 74 \& 7 | 13 18 23 29 35 42 58 76 \& 6 | 10 30 44 \& 5 | 7 11 15 20 32 46 62 80 \& 4 | 5 12 22 48 52 \& 3 | 3 6 14 24 33 55 \& 2 | 2 8 19 34 54 \& 1 | 1 4 9 16 25 36 49 64 81 \& Y=0 | \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 .Ve .PP The Y=1 bottom row is the perfect squares which are at i=j in the \f(CW\*(C`DiagonalsOctant\*(C'\fR and have gcd(i,j)=i thus becoming X=i,Y=1. .IX Xref "Square numbers" .PP .Vb 1 \& pairs_order => "diagonals_up" \& \& 9 | 25 29 39 45 58 65 \& 8 | 20 28 38 50 80 \& 7 | 16 19 23 27 32 37 63 78 \& 6 | 12 26 48 \& 5 | 9 11 14 17 35 46 59 74 \& 4 | 6 10 24 44 54 \& 3 | 4 5 15 22 34 51 \& 2 | 2 8 18 33 52 \& 1 | 1 3 7 13 21 31 43 57 73 \& Y=0 | \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 7 8 9 .Ve .PP N=1,2,4,6,9 etc in the X=1 column is the perfect squares k*k and the pronics k*(k+1) interleaved, also called the quarter-squares. N=2,5,10,17,etc on Y=X+1 above the leading diagonal are the squares+1, and N=3,8,15,24,etc below on Y=X\-1 below the diagonal are the squares\-1. .IX Xref "Square numbers Pronic numbers Quarter square numbers" .PP The \s-1GCD\s0 division moves points downwards and shears them across horizontally. The effect on diagonal lines of i,j points is as follows .PP .Vb 10 \& | 1 \& | 1 gcd=1 slope=\-1 \& | 1 \& | 1 \& | 1 \& | 1 \& | 1 \& | 1 \& | 1 \& | . gcd=2 slope=0 \& | . 2 \& | . 2 3 gcd=3 slope=1 \& | . 2 3 gcd=4 slope=2 \& | . 2 3 4 \& | . 3 4 5 gcd=5 slope=3 \& | . 4 5 \& | . 4 5 \& | . 5 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- .Ve .PP The line of \*(L"1\*(R"s is the diagonal with gcd=1 and thus X,Y=i,j unchanged. .PP The line of \*(L"2\*(R"s is when gcd=2 so X=(i+j)/2,Y=j/2. Since i+j=d is constant within the diagonal this makes X=d fixed, ie. vertical. .PP Then gcd=3 becomes X=(i+2j)/3 which slopes across by +1 for each i, or gcd=4 has X=(i+3j)/4 slope +2, etc. .PP Of course only some of the points in an i,j diagonal have a given gcd, but those which do are transformed this way. The effect is that for N up to a given diagonal row all the \*(L"*\*(R" points in the following are traversed, plus extras in wedge shaped arms out to the side. .PP .Vb 10 \& | * \& | * * up to a given diagonal points "*" \& | * * * all visited, plus some wedges out \& | * * * * to the right \& | * * * * * \& | * * * * * / \& | * * * * * / \-\- \& | * * * * * \-\- \& | * * * * *\-\- \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\- .Ve .PP In terms of the rationals X/Y the effect is that up to N=d^2 with diagonal d=2j the fractions enumerated are .PP .Vb 2 \& N=d^2 \& enumerates all num/den where num <= d and num+den <= 2*d .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::GcdRationals\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::GcdRationals\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::GcdRationals->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::GcdRationals\->new (pairs_order => $str)""" 4 .el .IP "\f(CW$path = Math::PlanePath::GcdRationals\->new (pairs_order => $str)\fR" 4 .IX Item "$path = Math::PlanePath::GcdRationals->new (pairs_order => $str)" .PD Create and return a new path object. The \f(CW\*(C`pairs_order\*(C'\fR option can be .Sp .Vb 4 \& "rows" (default) \& "rows_reverse" \& "diagonals_down" \& "diagonals_up" .Ve .SH "FORMULAS" .IX Header "FORMULAS" .SS "X,Y to N \*(-- Rows" .IX Subsection "X,Y to N Rows" The defining formula above for X,Y can be inverted to give i,j and N. This calculation doesn't notice if X,Y have a common factor, so a coprime(X,Y) test must be made separately if necessary (for \f(CW\*(C`xy_to_n()\*(C'\fR it is). .PP .Vb 1 \& X/Y = g\-1 + (i/g)/(j/g) .Ve .PP The g\-1 integer part is recovered by a division X divide Y, .PP .Vb 5 \& X = quot*Y + rem division by Y rounded towards 0 \& where 0 <= rem < Y \& unless Y=1 in which case use quot=X\-1, rem=1 \& g\-1 = quot \& g = quot+1 .Ve .PP The Y=1 special case can instead be left as the usual kind of division quot=X,rem=0, so 0<=rem (etc) .RE .PP .Vb 7 \& pairs_order="rows" (the default) \& A226314 X coordinate \& A054531 Y coordinate, being N/GCD(i,j) \& A000124 N in X=1 column, triangular+1 \& A050873 ceil(X/Y), gcd by rows \& A050873\-A023532 floor(X/Y) \& gcd by rows and subtract 1 unless i=j \& \& pairs_order="diagonals_down" \& A033638 N in X=1 column, quartersquares+1 and pronic+1 \& A000290 N in Y=1 row, perfect squares \& \& pairs_order="diagonals_up" \& A002620 N in X=1 column, squares and pronics \& A002061 N in Y=1 row, central polygonals (extra initial 1) \& A002522 N at Y=X+1 above leading diagonal, squares+1 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::DiagonalRationals, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns, Math::PlanePath::DiagonalsOctant .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .