Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::GcdRationals

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

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

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.

    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 rational is

    N = i + j*(j-1)/2     for upper triangle 1 <= i <= j
    gcd = GCD(i,j)
    rational = i/j + gcd-1

which means X=numerator Y=denominator are

    X = (i + j*(gcd-1))/gcd  = j + (i-j)/gcd
    Y = j/gcd

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.

    j=4  |  7  8  9 10
    j=3  |  4  5  6
    j=2  |  2  3
    j=1  |  1
         +-------------
          i=1  2  3  4

If 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, but 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. The integer part is GCD(i,j)-1 so rational = gcd-1 + i/j. For example

    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

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.

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.

    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

N=1,2,4,7,11,etc in the column at X=1 immediately follows each of those bottom row triangulars, ie. N+1.

    N in X=1 column = Y*(Y-1)/2 + 1

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 "..." 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  10

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.

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.

    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

If gcd odd then either j/gcd or j-1 is even, to take the "/2" 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 "/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=2

Of course knowing gcd or gcd/2 is a factor of N is only useful when that factor is 2 or more, so

    odd gcd >= 2                means gcd >= 3
    even gcd with gcd/2 >= 2    means gcd >= 4

    so N composite when gcd(i,j) >= 3

If 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 with 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  4

The X,Y 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  11

The triangular numbers, per "Triangular Numbers" 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.

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 are in the style of 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  4

The 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  9

The 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  9

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.

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 has 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 with diagonal d=2j the fractions enumerated are

    N=d^2
    enumerates all num/den where num <= d and num+den <= 2*d

FUNCTIONS ^

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_order option 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 necessary (for xy_to_n() it is).

    X/Y = g-1 + (i/g)/(j/g)

The g-1 integer part is recovered by a division X divide Y,

    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

The Y=1 special case can instead be left as the usual kind of division quot=X,rem=0, so 0<=rem<Y. This will give i=0 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 then it can be left rem=0.

Equating the denominators in the X/Y formula above gives j by

    Y = j/g          the definition above

    j = g*Y
      = (quot+1)*Y
    j = X+Y-rem      per the division X=quot*Y+rem

And equating the numerators gives i by

    X = (g-1)*Y + i/g     the definition above

    i = X*g - (g-1)*Y*g
      = X*g - quot*Y*g
      = X*g - (X-rem)*g     per the division X=quot*Y+rem
    i = rem*g
    i = rem*(quot+1)

Then N from i,j by the definition above

    N = i + j*(j-1)/2

For example X=11,Y=4 divides X/Y as 11=4*2+3 for quot=2,rem=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, not the remainder rem, by taking j=(quot+1)*Y instead of j=X+Y-rem, but usually a division operation gives the remainder at no extra cost, or a cost small enough that it's worth swapping a multiply for an add or two.

The gcd g can be recovered by rounding up in the division, instead of rounding down and then incrementing with g=quot+1.

   g = ceil(X/Y)
     = cquot for division X=cquot*Y - crem

But division in most programming languages is towards 0 or towards -infinity, not upwards towards +infinity.

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 + 1

The Y=1 case described above cannot be left to go through with rem=0 giving i=0 and j+1 since the reversal j-i+1 is then not correct. Either use rem=1 as described, or if not then compensate at the end,

    if r=0 then j -= 2            adjust
    Nrrev = j*(j+1)/2 - i + 1     same Nrrev as above

For example X=5,Y=1 is quot=5,rem=0 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)

    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

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, 2013, 2014 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.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

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