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.IX Title "Math::PlanePath::GcdRationals 3pm"
.TH Math::PlanePath::GcdRationals 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  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
<http://blog.computationalcomplexity.org/2004/03/counting\-rationals\-quickly.html>
.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<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.
.PP
Equating the denominators in the X/Y formula above gives j by
.PP
.Vb 1
\&    Y = j/g          the definition above
\&
\&    j = g*Y
\&      = (quot+1)*Y
\&    j = X+Y\-rem      per the division X=quot*Y+rem
.Ve
.PP
And equating the numerators gives i by
.PP
.Vb 1
\&    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)
.Ve
.PP
Then N from i,j by the definition above
.PP
.Vb 1
\&    N = i + j*(j\-1)/2
.Ve
.PP
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).
.PP
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.
.PP
The gcd g can be recovered by rounding up in the division, instead of
rounding down and then incrementing with g=quot+1.
.PP
.Vb 2
\&   g = ceil(X/Y)
\&     = cquot for division X=cquot*Y \- crem
.Ve
.PP
But division in most programming languages is towards 0 or towards
\&\-infinity, not upwards towards +infinity.
.SS "X,Y to N \*(-- Rows Reverse"
.IX Subsection "X,Y to N Rows Reverse"
For pairs_order=\*(L"rows_reverse\*(R", the horizontal i is reversed to j\-i+1.  This
can be worked into the triangular part of the N formula as
.PP
.Vb 2
\&    Nrrev = (j\-i+1) + j*(j\-1)/2        for 1<=i<=j
\&          = j*(j+1)/2 \- i + 1
.Ve
.PP
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,
.PP
.Vb 2
\&    if r=0 then j \-= 2            adjust
\&    Nrrev = j*(j+1)/2 \- i + 1     same Nrrev as above
.Ve
.PP
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
\&\*(L"Rows Reverse\*(R" above).
.SH "OEIS"
.IX Header "OEIS"
This enumeration of rationals is in Sloane's Online Encyclopedia of Integer
Sequences in the following forms
.Sp
.RS 4
<http://oeis.org/A054531> (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"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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.
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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.
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You should have received a copy of the \s-1GNU\s0 General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.