NAME
Math::PlanePath::GcdRationals -- rationals by triangular GCD
SYNOPSIS
use Math::PlanePath::GcdRationals;
my $path = Math::PlanePath::GcdRationals->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This path enumerates X/Y rationals using a method by Lance Fortnow taking a greatest common divisor out of a triangular position.
http://blog.computationalcomplexity.org/2004/03/counting-rationals-quickly.html
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 13The attraction of this approach is that it's both efficient to calculate and it traverses blocks of X/Y rationals with a modest range of N values, roughly a square N=2*max(num,den)^2 in the default rows style.
The mapping from N to rational is
N = i + j*(j-1)/2 for upper triangle 1 <= i <= j
gcd = GCD(i,j)
rational = i/j + gcd-1which means
X = (i + j*(gcd-1)) / gcd
Y = j/gcdThe i,j position is a numbering of points above the X=Y diagonal by rows (in the style of PyramidRows step=1).
j=4 | 7 8 9 10
j=3 | 4 5 6
j=2 | 2 3
j=1 | 1
+-------------
i=1 2 3 4If GCD(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, skipping positions where i,j have a common factor.
The skipped positions where i,j have a common factor become rationals X/Y>1, ie. below the X=Y diagonal. GCD(i,j)-1 is the integer part so rational = (gcd-1) + i/j. For example N=51 is at i=6,j=10 by rows and that i,j has common factor gcd(6,10)=2 so becomes rational R = (2-1) + 6/10 = 1+3/5 = 8/5, ie. X=8,Y=5.
Triangular Numbers
The bottom row Y=1 is the triangular numbers N=1,3,6,10,etc, k*(k-1)/2. Such an N is at i=k,j=k and thus gcd(i,j)=k which divides out to Y=1.
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 rowN=1,2,4,7,11,etc in the vertical at X=1 immediately following those triangulars on the bottom row, ie.
N on X=1 column = Y*(Y-1)/2 + 1Primes
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 "..." marking the X=2*Y line.
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 10Values 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. Other X=2*k,Y=k are not an X/Y rational in least terms due to common factor k.
This pattern of primes and composites occurs because N is a multiple of gcd(i,j) when gcd odd, or a multiple of gcd/2 when gcd even.
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 evenIf gcd odd then either j/gcd or j-1 is even, taking the "/2" factor. If gcd even then only gcd/2 can come out as a factor since the full gcd might leave both j/gcd and j-1 odd and so the "/2" not an integer. That happens for example to N=70
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=2Of course knowing gcd or gcd/2 is a factor is only useful when that factor is 2 or more, so
odd gcd with gcd >= 2 means gcd >= 3
even gcd with gcd/2 >= 2 means gcd >= 4
so N composite when gcd(i,j) >= 3If gcd<3 then the "factor" coming out is only 1 and says nothing about whether N is prime or composite. There are both prime and composite N for gcd<3, as can be seen among the values above the X=2*Y line in the table above.
Rows Reverse
Option pairs_order => "rows_reverse" reverses the order of points within the rows of i,j pairs,
j=4 | 10 9 8 7
j=3 | 6 5 4
j=2 | 3 2
j=1 | 1
+------------
i=1 2 3 4The point numbering becomes
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 11The triangular numbers per "Triangular Numbers" are now in the X=1 column, ie. at the left rather than the bottom. The Y=1 bottom row is the next after each triangular, ie. T(X)+1.
Diagonals
Option pairs_order => "diagonals_down" takes the i,j pairs by diagonals down from the Y axis. pairs_order => "diagonals_up" likewise but upwards from the X=Y centre up to the Y axis. (These numberings as per Math::PlanePath::DiagonalsOctant.)
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 4The resulting path becomes
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 9The Y=1 bottom row is the perfect squares which are at i=j in the DiagonalsOctant and have gcd(i,j)=i thus becoming X=i,Y=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 9N=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.
The GCD division moves points downwards and shears them across horizontally. The effect on diagonal lines of i,j points is as follows
| 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
+-------------------------------The line of "1"s is the diagonal with gcd=1 and thus X,Y=i,j unchanged.
The line of "2"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.
Then gcd=3 becomes X=(i+2j)/3 which slopes across by +1 for each i, or gcd=4 X=(i+3j)/4 slope +2, etc.
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 "*" points in the following are traversed, plus extras in wedge shaped arms out to the side.
| *
| * * up to a given diagonal points "*"
| * * * all visited, plus some wedges out
| * * * * to the right
| * * * * *
| * * * * * /
| * * * * * / --
| * * * * * --
| * * * * *--
+--------------In terms of the rationals X/Y the effect is that up to N=d^2 diagonal d=2j the fractions enumerated are
N=d^2
enumerates all num/den where num <= d and num+den <= 2*dFUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::GcdRationals->new ()$path = Math::PlanePath::GcdRationals->new (pairs_order => $str)-
Create and return a new path object. The
pairs_orderoption can be"rows" (default) "rows_reverse" "diagonals_down" "diagonals_up"
FORMULAS
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 desired.
X/Y = g-1 + (i/g)/(j/g)The g-1 integer part is recovered by a division
X = p*Y + r
where 0<=r<Y, unless Y=1 in which case use r=1, p=X-1
g-1 = p
g = p+1The Y=1 special case can instead be left as the usual kind of division r=0,p=X, so 0<=r<Y. This ends up i=0 below which is outside the intended 1<=i<=j range, but j is 1 bigger and the combination still gives the correct N. It's as if the i=g,j=g point at the end of a row is moved to i=0,j=g+1 just before the start of the next row. If only N is of interest not the i,j as such then it can be left r=0.
Equating the denominators in the X/Y formula above gives j
Y = j/g
j = g*Y
= (p+1)*Y
j = X+Y-r per the division X=p*Y+rAnd the numerators give i
X = (g-1)*Y + i/g
i = X*g - (g-1)*Y*g
= X*g - p*Y*g
= X*g - (X-r)*g per the division X=p*Y+r
i = r*g
i = r*(p+1)Then N from i,j by the definition
N = i + j*(j-1)/2For example X=11,Y=4 divides as 11=2*4+3 for p=2,r=3 so i=3*(2+1)=9 j=11+4-3=12 and so N=9+12*11/2=75 (as shown in the first table above).
It's possible to use only the quotient p by taking j=(p+1)*Y instead of j=X+Y-r, but usually a division operation gives the remainder at no extra cost, or a cost small enough that it's worth it to swap a multiply for an add or two.
X,Y to N -- Rows Reverse
For pairs_order="rows_reverse", the horizontal i is reversed to j-i+1. This can be worked into the triangular part of the N formula as
Nrrev = (j-i+1) + j*(j-1)/2 for 1<=i<=j
= j*(j+1)/2 - i + 1The Y=1 case described above cannot be left to go through with r=0 giving i=0 and j+1 since the reversal j-i+1 is then not correct. Either use r=1 as described, or if not then compensate at the end,
if r=0 then j += 2
Nrrev = j*(j+1)/2 - i + 1For example X=5,Y=1 is r=0,p=5 gives i=0*(5+1)=0 j=5+1-0=6. Without adjustment it would be Nrrev=6*7/2-0+1=22 which is wrong. But adjusting j-=2 so that j=6-2=4 gives the desired Nrrev=4*5/2-0+1=11 (per the table in "Rows Reverse" above).
OEIS
This enumeration of rationals is in Sloane's Online Encyclopedia of Integer Sequences in the following forms
http://oeis.org/A054531 (etc)
A054531 - Y coordinate, being N/GCD(i,j)SEE ALSO
Math::PlanePath, Math::PlanePath::DiagonalRationals, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns, Math::PlanePath::DiagonalsOctant
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2011, 2012 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
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