Math::PlanePath::FactorRationals -- rationals by prime powers
use Math::PlanePath::FactorRationals; my $path = Math::PlanePath::FactorRationals->new; my ($x, $y) = $path->n_to_xy (123);
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.
Option factor_coding => "odd/even" changes the f() mapping to numerator exponents as odd numbers and denominator exponents as even.
factor_coding => "odd/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.
Option factor_coding => "negabinary" changes the f() mapping to negabinary as suggested in
factor_coding => "negabinary"
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
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
factor_coding => "revbinary"
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.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::FactorRationals->new ()
$path = Math::PlanePath::FactorRationals->new (factor_coding => $str)
Create and return a new path object. factor_coding can be
factor_coding
"even/odd" (the default) "odd/even" "negabinary" "revbinary"
($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.
$n
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.
$x,$y
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.
$y
$x
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.
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
Math::PlanePath, Math::PlanePath::GcdRationals, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns
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.
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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