Kevin Ryde > Math-PlanePath > Math::PlanePath::FactorRationals

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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

Kevin McCrimmon, "Enumeration of the Positive Rationals", American Math. Monthly, Nov 1960, page 868. http://www.jstor.org/stable/2309448

Gerald Freilich, "A Denumerability Formula for the Rationals", American Math. Monthly, Nov 1965, pages 1013-1014. http://www.jstor.org/stable/2313350

Yoram Sagher, "Counting the rationals", American Math. Monthly, Nov 1989, page 823. http://www.jstor.org/stable/2324846

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

David M. Bradley, "Counting the Positive Rationals: A Brief Survey", http://arxiv.org/abs/math/0509025

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

http://oeis.org/A071974 (etc)

    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,

Ivan Niven, "Note on a paper by L. S. Johnston", American Math. Monthly, volume 55, number 6, June-July 1948, page 358. http://www.jstor.org/stable/2304962

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 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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