Math::PlanePath::GcdRationals -- rationals by prime factorization
use Math::PlanePath::GcdRationals; my $path = Math::PlanePath::GcdRationals->new; my ($x, $y) = $path->n_to_xy (123);
This path enumerates rationals X/Y using a method by Lance Fortnow taking a GCD 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 13
The mapping from N to X/Y is
N = i + j*(j-1)/2 upper triangle 1 <= i <= j gcd = GCD(i,j) rational = i/j + gcd-1 X = (i + j*(gcd-1)) / gcd Y = j/gcd
The i,j position is a numbering of points above the X=Y diagonal by rows,
j=4 7 8 9 10 j=3 4 5 6 j=2 2 3 j=1 1 i=1 2 3 4
When gcd=1, X/Y is simply X=i,Y=j. So the fractions X/Y < 1 above the X=Y diagonal are numbered by rows, ie. increasing numerator, skipping positions where X,Y have a common factor.
The skipped positions where i,j have a common factor become rationals R>1 below the X=Y diagonal. gcd(i,j)-1 is made the integer part in R = i/j+(gcd-1). For example N=51 is at i=6,j=10 by rows, but they have common factor gcd=2 so R = 6/10+(2-1) = 3/5+1 = 8/5, ie. X=8,Y=5.
The bottom row Y=1 is the triangular numbers N=1,3,6,10,etc, k*(k-1)/2. Such an N has i=k,j=k and thus gcd(i,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 row
All prime N values are above the sloping line X=2*Y. There's composites both above and below, but the primes are all above.
Here's the table with "..." marking the X=2*Y line. Only X=2,Y=1 is exactly on the line (which is prime N=3 as it happens) because X=2*k,Y=k is not an X/Y rational in least terms (it has common factor k). Values below X=2*Y such as 39 and 42 are all composites, values above like 19 and 30 are either prime or composite.
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 composites | .... below 2 | 2 8 .... 18 32 50 | .... 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
This occurs because N is a multiple of gcd(i,j) when the 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 even
For gcd odd either j/gcd or j-1 is even to take the "/2". When gcd is even 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 with N=70
N = 70 i = 4, j = 12 for 4 + 12*11/2 = 70 gcd = 4 but N is not a multiple of 4, only of 4/2=2
Of course the formula only shows N is composite when odd gcd>=3 or even gcd/2>=2, so gcd>=3, since otherwise the factor coming out is only 1. When gcd<3 there are in fact both prime and composite N, the values above the X=2*Y line in the sample above.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::GcdRationals->new ()
Create and return a new path object.
The defining formula above can be reversed
X/Y = i/j + g-1 g-1 = floor(X/Y) Y = j/g X = ((g-1)*j + i)/g j = Y*g i = X*g - (g-1)*Y*g N = i + j*(j-1)/2
So
N = g * ( X + Y*((Y-2)*g + 1)/2 ) with g = floor(X/Y) + 1
Either Y or (Y-2)*g+1 is even to take the /2. If Y is odd then g must be odd which makes (Y-2)*g odd and (Y-2)*g+1 even.
Y*g is the next multiple of Y which is strictly greater than X. It can be formed from the floor(X/Y) division
X = Y*q + r division g = q+1 Y*g = Y*(q+1) = X+Y-r = X+Y-r
Using X+Y-r instead of Y*g in the N formula might swap a multiply for an add or subtract if you get the remainder for free with the X/Y division.
An over-estimate of the N range can be calculated just from the X,Y to N formula above ignoring which X,Y points are coprime and thus actually should have N values.
Within a row N increases with increasing X, so for a rectangle the minimum is in the left column and the maximum in the right column.
Within a column N values increase until reaching the end of a "g" wedge, then drop down a bit. So the maximum is either the top-right corner or the top of the next lower wedge, ie. smaller y but bigger g. And conversely the minimum is either the bottom right, or the start of the next higher wedge, ie. smaller g but bigger y. (That's right is it?)
This enumeration of rationals is in Sloane's Online Encyclopedia of Integer Sequences in the following forms
http://oeis.org/A054531 (etc) A054531 - denominators, Y coordinate
Math::PlanePath, Math::PlanePath::DiagonalRationals, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012 Kevin Ryde
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