/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::FactorRationals

NAME

Math::PlanePath::FactorRationals -- rationals by prime powers

SYNOPSIS

use Math::PlanePath::FactorRationals;
my $path = Math::PlanePath::FactorRationals->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path enumerates rationals X/Y with no common factor, based on the prime powers in numerator and denominator, as per

The result is

15  |      15   60       240            735  960           1815
14  |      14       126       350                1134      1694
13  |      13   52  117  208  325  468  637  832 1053 1300 1573
12  |      24                 600      1176                2904
11  |      11   44   99  176  275  396  539  704  891 1100
10  |      10        90                 490       810      1210
 9  |      27  108       432  675      1323 1728      2700 3267
 8  |      32       288       800      1568      2592      3872
 7  |       7   28   63  112  175  252       448  567  700  847
 6  |       6                 150       294                 726
 5  |       5   20   45   80       180  245  320  405       605
 4  |       8        72       200       392       648       968
 3  |       3   12        48   75       147  192       300  363
 2  |       2        18        50        98       162       242
 1  |       1    4    9   16   25   36   49   64   81  100  121
Y=0 |
     ----------------------------------------------------------
      X=0   1    2    3    4    5    6    7    8    9   10   11

A given fraction X/Y with no common factor has a prime factorization

X/Y = p1^e1 * p2^e2 * ...

The exponents e[i] are positive, negative or zero, being positive when the prime is in the numerator or negative when in the denominator. Those exponents are represented in an integer N by mapping the exponents to non-negative,

N = p1^f(e1) * p2^f(e2) * ...

f(e) = 2*e      if e >= 0
     = 1-2*e    if e < 0

f(e)      e
---      ---
 0        0
 1       -1
 2        1
 3       -2
 4        2

For example

X/Y = 125/7 = 5^3 * 7^(-1)
encoded as N = 5^(2*3) * 7^(1-2*(-1)) = 5^6 * 7^1 = 5359375

N=3   ->  3^-1 = 1/3
N=9   ->  3^1  = 3/1
N=27  ->  3^-2 = 1/9
N=81  ->  3^2  = 9/1

The effect is to distinguish prime factors of the numerator or denominator by odd or even exponents of those primes in N. Since X and Y have no common factor a given prime appears in one and not the other. The oddness or evenness of the p^f() exponent in N can then encode which of the two X or Y it came from.

The exponent f(e) in N has term 2*e in both cases, but the exponents from Y are reduced by 1. This can be expressed in the following form. Going from X,Y to N doesn't need to factorize X, only Y.

         X^2 * Y^2
N = --------------------
    distinct primes in Y

N=1,2,3,8,5,6,etc in the column X=1 is integers with odd powers of prime factors. These are the fractions 1/Y so the exponents of the primes are all negative and thus all exponents in N are odd.

N=1,4,9,16,etc in row Y=1 are the perfect squares. That row is the integers X/1 so the exponents are all positive and thus in N become 2*e, giving simply N=X^2.

Odd/Even

Option factor_coding => "odd/even" changes the f() mapping to numerator exponents as odd numbers and denominator exponents as even.

f(e) = 2*e-1    if e > 0
     = -2*e     if e <= 0

The effect is simply to transpose X<->Y.

"odd/even" is the form given by Kevin McCrimmon and Gerald Freilich. The default "even/odd" is the form given by Yoram Sagher.

Negabinary

Option factor_coding => "negabinary" changes the f() mapping to negabinary as suggested in

This coding is not as compact as odd/even and tends to make bigger N values,

13  |    2197   4394   6591 140608  10985  13182  15379 281216
12  |     108                         540           756
11  |    1331   2662   3993  85184   6655   7986   9317 170368
10  |    1000          3000                        7000
 9  |       9     18           576     45            63   1152
 8  |    8192         24576         40960         57344
 7  |     343    686   1029  21952   1715   2058         43904
 6  |     216                        1080          1512
 5  |     125    250    375   8000           750    875  16000
 4  |       4            12            20            28
 3  |      27     54          1728    135           189   3456
 2  |       8            24            40            56
 1  |       1      2      3     64      5      6      7    128
Y=0 |
     ----------------------------------------------------------
      X=0   1      2      3      4      5      6      7      8

Reversing Binary

Option factor_coding => "revbinary" changes the f() mapping to "reversing binary" where a given integer is represented as a sum of powers 2^k with alternating signs

e = 2^k1 - 2^k2 + 2^k3 - ...           0 <= k1 < k2 < k3

f(e)      e
---      ---
 0        0
 1        1
 2        2
 3       -1
 4        4
 5       -3
 6       -2
 7        3

This representation is per Knuth volume 2 section 4.1 exercise 27. The exercise there is to show all integers can be represented this way.

 9  |     729  1458        2916  3645        5103 93312        7290
 8  |      32          96         160         224         288
 7  |     343   686  1029  1372  1715  2058       43904  3087  3430
 6  |     216                    1080        1512
 5  |     125   250   375   500         750   875 16000  1125
 4  |      64         192         320         448         576
 3  |      27    54         108   135         189  3456         270
 2  |       8          24          40          56          72
 1  |       1     2     3     4     5     6     7   128     9    10
Y=0 |
     ---------------------------------------------------------------
      X=0   1     2     3     4     5     6     7     8     9    10

The X axis begins with the integers 1 to 7 because f(1)=1 and f(2)=2 so N=X until X has a prime p^3 or higher power. The first such is X=8=2^3 which is f(7)=3 so N=2^7=128.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::FactorRationals->new ()
$path = Math::PlanePath::FactorRationals->new (factor_coding => $str)

Create and return a new path object. factor_coding can be

"even/odd"    (the default)
"odd/even"
"negabinary"
"revbinary"
($x,$y) = $path->n_to_xy ($n)

Return X,Y coordinates of point $n on the path. If there's no point $n then the return is an empty list.

This depends on factorizing $n and in the current code there's a hard limit on the amount of factorizing attempted. If $n is too big then the return is an empty list.

$n = $path->xy_to_n ($x,$y)

Return the N point number for coordinates $x,$y. If there's nothing at $x,$y, such as when they have a common factor, then return undef.

This depends on factorizing $y, or factorizing both $x and $y for negabinary or revbinary. In the current code there's a hard limit on the amount of factorizing attempted. If the coordinates are too big then the return is undef.

The current factorizing limits handle anything up to 2^32, and above that numbers comprised of small factors. But big numbers with big factors are not handled. Is this a good idea? For large inputs there's no merit in disappearing into a nearly-infinite loop. Perhaps the limits could be configurable and/or some advanced factoring modules attempted for a while if/when available.

OEIS

This enumeration of the rationals is in Sloane's Online Encyclopedia of Integer Sequences in the following forms

A071974   X coordinate, numerators
A071975   Y coordinate, denominators
A019554   X*Y product
A102631   N in column X=1, n^2/squarefreekernel(n)
A072345   X and Y at N=2^k, being alternately 1 and 2^k

A011262   permutation N at transpose Y/X (exponents mangle odd<->even)

A060837   permutation DiagonalRationals -> FactorRationals
A071970   permutation RationalsTree CW -> FactorRationals

The last A071970 is rationals taken in order of the Stern diatomic sequence stern[i]/stern[i+1] which is the Calkin-Wilf tree rows ("Calkin-Wilf Tree" in Math::PlanePath::RationalsTree).

The negabinary representation is

A053985   index -> signed
A005351   signed positives -> index
A039724   signed positives -> index, in binary
A005352   signed negatives -> index

The reversing binary representation is

A065620   index -> signed
A065621   signed positives -> index
A048724   signed negatives -> index

SEE ALSO

Math::PlanePath, Math::PlanePath::GcdRationals, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns

Other Ways to Do It

Niven gives another prime factor based construction but encoding N by runs of 1-bits,

N is written in binary each 0-bit is considered a separator. The number of 1-bits between each

N = 11 0 0 111 0 11  binary
       | |     |
     2  0   3    2   f(e) = run lengths of 1-bits
    -1  0  +2   -1   e exponent by "odd/even" style

X/Y = 2^(-1) * 3^(+2) * 5^0 * 7^(-1)

Kevin McCrimmon's note begins with a further possible encoding for N where the prime powers from numerator are spread out to primes p[2i+1] and with 0 powers sending a p[2i] power to the denominator. In this form the primes from X and Y spread out to different primes rather than different exponents.

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/>.