Egyptian Fractions

Fast algorithm for representing rational numbers as egyptian fractions.

Properties

• suitable for very large inputs
• returns rather small denominators

Usage

Egyptian Fractions

Usage: egypt [OPTIONS] [NUMERATOR] [DENOMINATOR]

Arguments:
  [NUMERATOR]    [default: 1]
  [DENOMINATOR]  [default: 1]

Options:
  -r, --reverse        Reverse merge strategy
  -m, --merge          Extra O(n^2) merge step possibly reducing number of terms
      --raw            Output minimal number of raw quadruplets (aka symbolic sums)
      --bisect         Output raw quadruplets bisected according to --limit
  -s, --silent         No output
      --batch          Batch mode (expects numerator and denominator on each line of stdin)
  -l, --limit <LIMIT>  Maximum number of terms for breaking large symbolic sums [default: 8]
  -h, --help           Print help
  -V, --version        Print version

Performance

$ time ./egypt -s '2 9689 ^ 1 -' '2 9941 ^ 1 -'

real    0m0.566s
user    0m0.528s
sys     0m0.036s

$ time ./egypt -s 162259276829213363391578010288127 170141183460469231731687303715884105727

real    0m0.002s
user    0m0.001s
sys     0m0.001s

$ time ./egypt --limit 2 999999 1000000
1       2
1       4
1       8
1       16
1       33
1       50
1       83
1       12450
1       32912
1       49368
1       90387
1       285716
1       571432
1       1999996
1       685610442
1       2057423870
1       11904714285
1       83332333335
1       499999000000

real    0m0.002s
user    0m0.000s
sys     0m0.002s

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Examples

7 / 19

• Wolfram|Alpha
  • 1 / 3 + 1 / 29 + 1 / 1653
• egypt --merge --limit 19 7 19
  • 1 / 3 + 1 / 33 + 1 / 209

2023 / 2024

• Wolfram|Alpha
  • 1 / 2 + 1 / 3 + 1 / 7 + 1 / 43 + 1 / 16768 + 1 / 766160103 + 1 / 978335504948790912
• egypt --merge --limit 2023 2023 2024
  • 1 / 2 + 1 / 3 + 1 / 8 + 1 / 33 + 1 / 92
• egypt --reverse --merge --limit 2023 2023 2024
  • 1 / 2 + 1 / 3 + 1 / 7 + 1 / 43 + 1 / 18447 + 1 / 184184
• egypt --limit 2 2023 2024
  • 1 / 2 + 1 / 4 + 1 / 8 + 1 / 11 + 1 / 33 + 1 / 674 + 1 / 899 + 1 / 2442 + 1 / 4044 + 1 / 24938 + 1 / 2046264 + 1 / 2423704

Note

• returns rather small denominators

When using legacy configuration egypt --merge --limit <LIMIT> <NUMERATOR> <DENOMINATOR>, where LIMIT >= DENOMINATOR - 1, largest denominator factor should not be greater than original denominator. Fast default limit is however 2, which means that bisecting large symbolic sums can introduce bigger denominators. Moreover, --limit argument is itself limited by usize, as opposed to other BigInt inputs.

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Illustrations

• Lissajous Curves: https://mathworld.wolfram.com/LissajousCurve.html

Approximating rational numbers

by different rational numbers with denominator coprime to the original denominator

7 / 11

7 / 11 errors