P-adic numbers

A collection of tools for p-adic numbers in Rust.

This includes a p-adic type and a rational type.

P-adic notation for the expansion is currently left to right.

Status

This library is currently in development and might be unstable.

Usage

Add this to your Cargo.toml:

toml [dependencies] padic = "0.1.5"

rust use padic::Ratio; let ratio = Ratio::new(2, 5); let padic = r.to_padic(3, 12); assert_eq!(padic.valuation, 0); assert_eq!(padic.expansion, vec![1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1]); assert_eq!(padic.to_string(), "... 1 2 1 0 1 2 1 0 1 2 1 1"); assert_eq!(padic.cycle(), [1, 2, 1, 0]);

Helpers functions

• Prime factors with multiplicity (a: i64 / b: i64) -> Vec<(prime: u64, exp: u64)>
• Greatest common divisor (Stein's algorithm)
• Modular multiplicative inverse
• Double cursor window cycle detection for repeating digits in p-adic expansion

Resources

• https://en.wikipedia.org/wiki/P-adic_number
• https://www.cut-the-knot.org/blue/p-adicExpansion.shtml

TODOs

Ratio

• [x] Extract sign information to transform ratio into a tuple of unsigned integer variables
• [x] Reduce ratio to lowest terms using GCD (Stein's algorithm)
• [x] Basic arithmetic operations for rational numbers
• [x] Modular multiplicative inverse
• [ ] Implement extended greatest common divisor to extract Bezout coefficients

P-adic

• [x] Prime decomposition returning vector of (prime, exponent) tuples.
• [x] P-adic valuation of rational number
• [x] P-adic norm of rational number
• [x] P-adic expansion of rational number with given precision
• [x] P-adic string representation with given precision and given valuation
• [x] Cyclic detection in p-adic expansion (Sliding window algorithm)
• [ ] P-adic arithmetic operations
• [ ] Convert p-adic expansion into rational number

Bugs / Features

• [ ] If the valuation is larger than the precision, the expansion is not correct
• [ ] If the precision is lower than the cycle length, the cycle is not detected

License

MIT