Provided by: libmath-planepath-perl_129-1_all bug

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, 2015, 2016, 2017, 2018, 2019, 2020 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/>.