https://github.com/popojan/egypt
egyptian fractions with small denominators
https://github.com/popojan/egypt
egyptian-fractions rational-numbers rationalization sqrt
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egyptian fractions with small denominators
- Host: GitHub
- URL: https://github.com/popojan/egypt
- Owner: popojan
- Created: 2023-06-28T21:23:43.000Z (almost 3 years ago)
- Default Branch: main
- Last Pushed: 2025-12-05T18:18:56.000Z (6 months ago)
- Last Synced: 2025-12-09T06:44:20.994Z (6 months ago)
- Topics: egyptian-fractions, rational-numbers, rationalization, sqrt
- Language: Rust
- Homepage:
- Size: 1.75 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# 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 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 0m0.345s
user 0m0.296s
sys 0m0.047s
```
```
$ 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
```
---
## 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
## Irrational / Transcendental Numbers
Supports RPN expressions with constants: `pi`, `e`, `phi` (golden ratio), `sqrt2`, `gamma` (Euler-Mascheroni).
```bash
# Pi/4 as Egypt fractions (raw symbolic tuples)
$ egypt pi 4 --raw -p 64
1 1 1 3
4 5 1 1
9 14 1 15
219 452 1 72
32763 33215 1 9
331698 364913 1 17
6535219 6900132 1 2
20335483 27235615 1 5
```
Each tuple `(u, v, i, j)` represents: `sum_{k=i}^{j} 1/((u-v+vk)(u+vk))`
The `-p` flag controls precision in bits (default 256). Higher precision = more CF terms = more tuples.
```bash
# Golden ratio
$ egypt phi 1 --raw -p 64 | head -5
1 0 0 0
1 1 1 1
1 2 1 1
2 3 1 1
3 5 1 1
```
Note: For irrationals, output represents a finite-precision rational approximation.
Early tuples are stable convergents of the true constant; later ones may diverge.
### Pell Equation Solver
Find solutions to Pell equation p² - D·q² = ±1 using `--pell` flag:
```bash
# sqrt(13): fundamental solution 649² - 13·180² = 1
$ egypt "13 sqrt" 1 --pell -p 64
q p norm
1 3 -4
...
180 649 1
# Fundamental solution (norm=1): p=649, q=180
# Cattle problem (D=4729494)
$ egypt "4729494 sqrt" 1 --pell -p 512 | tail -1
50549485234315033074477819735540408986340 109931986732829734979866232821433543901088049 1
```
Egypt tuples encode CF convergent denominators directly, making Pell extraction
a byproduct of the representation. Precision (`-p`) must be sufficient for the
CF period length; use higher values for larger D.
## Note
> * returns rather small denominators
>
When using legacy configuration `egypt --merge --limit `, 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.
## Relation to Continued Fractions
```
<< "wl/Egypt.wl"
compare[Rational[p_, q_]] := {
Total /@ Partition[ Differences @ Convergents[ p/q ], 2],
ReleaseHold @ EgyptianFractions[ p/q , Method -> "Expression"]
}
```
**Theorem**: Egypt values equal paired differences of continued fraction convergents.
This explains the monotonicity property: paired CF differences cancel the alternating sign pattern.
---
## Theory & Related Work
For theoretical background on the symbolic telescoping representation:
- **Paper**: [Egyptian Fractions via Modular Inverse: Symbolic Telescoping Representation](https://github.com/popojan/orbit/blob/main/docs/papers/egyptian-fractions-telescoping.tex)
- **Wolfram implementation**: [`Orbit/Kernel/EgyptianFractions.wl`](https://github.com/popojan/orbit/blob/main/Orbit/Kernel/EgyptianFractions.wl)
- **CF-Egypt Bijection (Proven Dec 2025)**: [Session documentation](https://github.com/popojan/orbit/blob/main/docs/sessions/2025-12-10-cf-egypt-equivalence/README.md)
- **γ-Egypt Simplification**: [Characterization theorems](https://github.com/popojan/orbit/blob/main/docs/sessions/2025-12-10-cf-egypt-equivalence/gamma-egypt-simplification.md)
The symbolic representation `{u, v, i, j}` compresses consecutive unit fractions into
telescoping sums with closed form: `(j-i+1) / ((u-v+vi)(u+vj))`.
This reduces complexity from O(numerator) expanded fractions to O(log denominator) symbolic tuples.
### CF-Egypt Bijection (✅ Proven)
For q = a/b with CF [0; a₁, a₂, ..., aₙ] and convergent denominators {q₀=1, q₁, ..., qₙ=b}:
| Case | u_k | v_k | j_k |
|------|-----|-----|-----|
| Regular (k < ⌈n/2⌉ or n even) | q_{2k-2} | q_{2k-1} | a_{2k} |
| Last tuple, odd CF | q_{n-1} | q_n - q_{n-1} | 1 |
**Key insight:** XGCD quotients ARE CF coefficients, enabling single-pass computation.
---
## 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
