Adic number math

Adic numbers

p-adic numbers are an alternate number system to the reals. This system is p-periodic and hierarchical. It is used throughout number theory, but it not well-known outside of pure math. This crate is partially an attempt to change that.

Links

• crates.io/adic
• docs/adic
• gitlab/adic
• adicmath.com - our site, to learn about and play with adic numbers
• crates.io/adic-shape - crate for visualizations and charts of adic numbers
• wiki/P-adic_number - wikipedia for p-adic numbers

Motivation

Adic numbers can represent any rational number as well as many numbers between them, just like the real numbers. They can be represented similarly to the reals as infinite digital expansions. Except where the reals have a finite number of digits to the left of the decimal and possibly infinite to the right (1.414...), the adics have finite digits to the right and infinite to the left (...4132.13).

assert_eq!("2314._5", uadic!(5, [4, 1, 3, 2]).to_string());
assert_eq!("(4)11._5", iadic_neg!(5, [1, 1]).to_string());
assert_eq!("(233)4._5", radic!(5, [4], [3, 3, 2]).to_string());
assert_eq!("...004123._5", zadic_approx!(5, 6, [3, 2, 1, 4]).to_string());
assert_eq!("...0041.23_5", qadic!(zadic_approx!(5, 6, [3, 2, 1, 4]), -2).to_string());
assert_eq!("...0ld._25", apow!(zadic_approx!(5, 6, [3, 2, 1, 4]), 2).to_string());

You might think this means they are "infinite" numbers, but they are not! The key difference is how a number's size is measured.

For a number, its "size" is its norm, its absolute value. In the reals, the size of 4 is 4, the size of -2.31 is 2.31, etc.

Each p-adic space is linked to a prime, p. In the p-adics, the size of a number is the inverse of how many powers of p are in it: |x| = |a/b * p^v| = p^(-v). So in the 5-adics, 1, 2, 3, and 4 are all size 1, while 5, 10, 15, and 20 are size 1/5. When you represent these numbers as digital expansions in base-p, the numbers further to the left are SMALLER, not bigger.

let one = uadic!(5, [1]);
let two = uadic!(5, [2]);
let three = uadic!(5, [3]);
let five = uadic!(5, [0, 1]);
let ten = uadic!(5, [0, 2]);
let twenty_five = uadic!(5, [0, 0, 1]);
let six_hundred_twenty_five = uadic!(5, [0, 0, 0, 0, 1]);
assert_eq!(Ratio::new(1, 1), one.norm());
assert_eq!(Ratio::new(1, 1), two.norm());
assert_eq!(Ratio::new(1, 1), three.norm());
assert_eq!(Ratio::new(1, 5), five.norm());
assert_eq!(Ratio::new(1, 5), ten.norm());
assert_eq!(Ratio::new(1, 25), twenty_five.norm());
assert_eq!(Ratio::new(1, 625), six_hundred_twenty_five.norm());

Adic numbers are used:

• to solve diophantine equations
• as fundamental examples of ultrametric spaces
• to form combined local/global structures, e.g. adeles and ideles
• in glassy physical systems, like in replica/cavity theory
• in tropical geometry

Crate

This crate handles adic numbers, arithmetic, and calculations.

Calculations:

• Adic arithmetic: Add, Sub, Mul, Div, Pow, Neg
• Rootfinding, through use of hensel lifting
• Idempotent computation for AdicComposite zero divisors

Adic number structs:

Related adic structs:

• AdicValuation - Valuation for adic numbers
• LazyDiv - Lazily calculate adic number division

Adic number traits:

• AdicNumber - A number with a prime and Add/Mul arithmetic
• SignedAdicNumber - An AdicNumber that includes Neg and Sub
• RationalAdicNumber - An AdicNumber that can represent rational numbers
• AdicInteger - An adic number without p-fractional digits
• AdicFraction - An adic number with p-fractional digits
• AdicSized - An adic number with valuation and norm
• AdicApproximate - An adic number with certainty and significance
• HasDigits - A structure with indexed digits

Polynomials:

• AdicPolynomial - Polynomial with adic integer coefficients
• AdicVariety - A collection of approximate AdicNumbers representing the roots of a polynomial

Divisible:

• Divisible - A trait for structures made of prime decompositions
• Prime - A prime number
• PrimePower - The power of a Prime
• Composite - A combination of PrimePowers
• Natural - Composite or zero (note: NOT a Divisible struct)

Example: calculate the two varieties for 7-adic sqrt(2) to 6 digits:

use adic::{uadic, zadic_variety, AdicInteger};
// Create the 7-adic number 2
let seven_adic_two = uadic!(7, [2]);
// Take the square root of seven_adic_two, to 6 "decimal places"
let sqrt_two_variety = seven_adic_two.nth_root(2, 6);
assert_eq!(Ok(zadic_variety!(7, 6, [
    [3, 1, 2, 6, 1, 2],
    [4, 5, 4, 0, 5, 4],
])), sqrt_two_variety);
let roots = sqrt_two_variety?.into_roots().collect::<Vec<_>>();
assert_eq!("...216213._7", roots[0].to_string());
assert_eq!("...450454._7", roots[1].to_string());

Example: 5-adic arithmetic

use adic::{uadic, radic, AdicInteger};
// 3 is a single digit (3) and no repeating digits
let three = radic![5, [3], []];
// -1/6 consists only of repeating ...040404.
let neg_one_sixth = radic![5, [], [4, 0]];
// 3 - 1/6 = 17/6 is two digits 12. and then repeating 04
let seventeen_sixth = three + neg_one_sixth;
assert_eq!(radic![5, [2, 1], [4, 0]], seventeen_sixth);
assert_eq!(uadic![5, [2, 1, 4, 0, 4, 0]], seventeen_sixth.truncation(6));

Example: 5-adic fourth roots of unity

use num::traits::Pow;
use adic::{roots_of_unity, zadic_approx, zadic_variety, AdicInteger};
// Every (odd) p-adic number space has p-1 roots of unity
let roots = roots_of_unity(5, 6)?;
assert_eq!(
    zadic_variety!(5, 6, [[1], [2, 1, 2, 1, 3, 4], [3, 3, 2, 3, 1, 0], [4, 4, 4, 4, 4, 4]]),
    roots
);
let approx_one = zadic_approx!(5, 6, [1]);
for root in roots.into_roots() {
    assert_eq!(approx_one, root.pow(4));
}

TODO

• QXAdic for a number from a finite extension of QAdic
• QCAdic for a number in the algebraic closure of QAdic
• CAdic, a "complex adic number", in the norm completion of QCAdic
• SAdic, a "spherically complete adic number", in the spherical completion of QCAdic/CAdic

License: MIT OR Apache-2.0