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use crate::number_theory::UnsigInt;
use crate::number_theory::util;
use crate::number_theory::primes_less_than::{
//sieve::Sieve,
PrimesLessThan};
pub struct Bernstein1988
{
n : u32,
f : u32
}
#[cfg(test)]
mod tests {
use super::Bernstein1988;
#[test]
fn recognizes_perfect_powers(){
use std::vec::IntoIter;
let mut iter : IntoIter<u32> =
vec![
625,729,841,961,1089,1225,1369,1521,1681,1849,2025,2209,2401,2601,2809,
3025,3249,3481,3721,3969,4225,4489,4761,5041,5329,5625,5929,6241,6561,6889,7225,7569,7921,8281,8649,9025,
9409,9801,10201,10609,11025,11449,11881,12321,12769,13225,13689,14161,14641,15129,15625,16129,16641,17161,
17689,18225,18769,19321,19881,20449,21025,21609,22201,22801,23409,24025,24649,25281, // Squared numbers
2401, // 7^4
3125, // 5^5
6561, // 3^8
1_771_561, // 11^6
].into_iter();
assert!(iter.all(|x| Bernstein1988::is_perfect_power(x)));
}
#[test]
fn recognizes_not_perfect_powers(){
use std::vec::IntoIter;
let mut iter : IntoIter<u32> =
vec![
627,731,843,963,1091,1227,1371,1523,1683,1851,2027,2211,2403,2603,2811,
3027,3251,3483,3723,3971,4227,4491,4763,5043,5331,5627,5931,6243,6563,6891,7227,7571,7923,8283,8651,9027,
9411,9803,10203,10611,11027,11451,11883,12323,12771,13227,13691,14163,14643,15131,15627,16131,16643,17163,
17691,18227,18771,19323,19883,20451,21027,21611,22203,22803,23411,24027,24651,25283, // Squared numbers + 2
2403, // 7^4 + 2
3127, // 5^5 + 2
6563, // 3^8 + 2
1_771_563 // 11^6 + 2
].into_iter();
assert!(iter.all(|x| !Bernstein1988::is_perfect_power(x)));
}
}
impl Bernstein1988 {
fn new(n :u32) -> Self {
Self{n, f : util::log2_floor(2*n)}
}
fn mul_2<U: UnsigInt>(b: u32, m: U, k: U) -> U {
(k * m) % (U::two().pow(b))
}
// Unique integer d between 0 and 2^b s.t
// (m = kd) % exp aka (m/k)%exp
fn div<U: UnsigInt>(exp: U, m: U, k: U, neg : bool) -> U {
if util::is_even(k) {panic!("expected k odd, {} given", k)}
let m_mod = m % exp ;
for d in num::range(U::zero(), exp-U::one()) {
let kd_mod = (k * d) % exp;
if m_mod == kd_mod {
if neg {
return exp - d
} else {
return d
}
}
}
if neg {
U::one()
} else {
exp - U::one()
}
}
// x**d mod m
// fn power(x : u32, k :u32, m :u32) -> u32{
// let mut x = x;
// let mut k = k;
// let mut p = 1u32;
// if x == 1u32 { return p }
// x %= m;
// while k>= 1{
// if k%2 == 0 {
// p = (p*x) % m
// }
// x = (x.pow(2u32)) % m;
// k /= 2;
// }
// p
// }
// Power up to b bits
// n**k mod 2**b
fn pow_2<U: UnsigInt>(b: u32, x: U, k: U) -> U {
util::powm(x, k, U::two().pow(b))
//Self::power(x, k, 2u32.pow(b))
}
// Algorithm P
// Algorithm C2
fn n_eq_x_pow_k(&self, n: u32, x: u32, k: u32) -> bool {
//println!("{} == {}^{}", n, x, k);
if x == 1 {
return n == 1;
}
let mut b = 1;
loop {
let exp = 2u32.pow(b);
let r = x.pow(k) % exp;
if n % exp != r {
return false;
}
if b >= self.f {
return r == x.pow(k);
}
b = (2 * b).min(self.f);
}
}
// Algorithm N2
// nroot_{2,b}(y,k)
fn nroot(b: u32, y: u32, k: u32) -> u32 {
if util::is_even(y) {panic!("expected y odd, {} given", y)}
if k == 2 {return Self::nroot_2(b, y)}
else if util::is_even(k) {panic!("expected k odd, {} given", k)}
//println!("b = {} y = {} k = {}", b, y ,k);
if b == 1 {
return 1;
}
let mut b = b;
let mut vec_b = Vec::with_capacity(util::log2_floor(b) as usize);
while b > 1{
vec_b.push(b);
if b%2 != 0 {
b += 1;
}
b /= 2;
//println!("In end: b = {}", b);
}
vec_b.reverse();
//println!("{:?}", vec_b);
let mut z = 1;
for b in vec_b{
let exp = 2u32.pow(b);
let r2 = Self::mul_2(b, z, k + 1);
//println!("r2 = {}", r2);
let r3 = Self::mul_2(b, y, Self::pow_2(b, z, k+1));
//println!("r3 = {}", r3);
z = if r2 < r3 {
Self::div(exp, r3 - r2, k, true)
} else {
Self::div(exp, r2 - r3, k, false)
};
//println!("z = {}", z)
}
z
}
// Algorithm S2
// nroot_{2,b}(y,2)
fn nroot_2(b: u32, y: u32) -> u32 {
//println!("b = {} y = {} k = {}", b, y ,2);
if util::is_even(y) {panic!("expected y odd, {} given", y)}
if b == 1 {
return if y % 4 == 1 {1} else {0};
}
if b == 2 {
return if y % 8 == 1 {1} else {0};
}
let mut b = b;
let mut vec_b = Vec::with_capacity(util::log2_floor(b) as usize);
while b > 2{
//println!("b = {}", b);
vec_b.push(b);
b = if b%2 != 0 {
(b+1)/2
} else {
1 + (b/2)
};
}
vec_b.reverse();
//println!("{:?}", vec_b);
if y % 8 != 1 {return 0}
let mut z = 1;
for b in vec_b {
let r2 = Self::mul_2(b+1, z, 3);
//println!("r2 = {}", r2);
let r3 = Self::mul_2(b+1, y, Self::pow_2(b+1, z, 3));
//println!("r3 = {}", r3);
let exp = 2u32.pow(b);
z = if r2 < r3 {
exp - (((r3-r2)/2)%exp)
} else {
((r2-r3)/2)%exp
};
//println!("z = {}", z);
if z == 0 {return 0}
}
z
}
// Algorithm K2
fn is_kth_power(&self, k: u32, y: u32) -> bool {
if util::is_even(self.n) {panic!("expected n odd, {} given", self.n)}
if util::is_even(k) && k != 2 {panic!("expected k odd or k=2, {} given", k)}
if util::is_even(y) {panic!("expected y odd, {} given", y)}
let b = util::div_ceiling(self.f, k);
//println!("{} == ?^{}", n, k);
let r : u32;
if k == 2 {
r = Self::nroot_2(b, y);
if r == 0 {return false}
} else {
r = Self::nroot(b, y, k);
}
//println!("r = {}", r);
if self.n_eq_x_pow_k(self.n, r, k) {
return true;
}
if k == 2 && self.n_eq_x_pow_k(self.n, 2u32.pow(b) - r, k){
return true;
}
false
}
// Algorithm X2
/// Checks whether `n` is a perfect power
/// # Examples
/// ```
/// use tnt_lib::Bernstein1988;
/// let mut pp = vec![
/// 121, // 11^2
/// 3125, // 5^5
/// 6561 // 3^8
/// ].into_iter();
/// assert!(pp.all(|x| Bernstein1988::is_perfect_power(x)));
///
/// let mut not_pp = vec![
/// 123, // 11^2 + 2
/// 3127, // 5^5 + 2
/// 6563 // 3^8 + 2
/// ].into_iter();
/// assert!(not_pp.all(|x| !Bernstein1988::is_perfect_power(x)));
/// ```
pub fn is_perfect_power(n : u32) -> bool {
use crate::number_theory::primes_less_than::sieve::Sieve;
if util::is_even(n as u32) {panic!("expected n odd, {} given", n)}
let test = Self::new(n);
let b = util::div_ceiling(test.f, 2);
let y = Self::nroot(b + 1, n as u32, 1);
//println!("f = {} b = {} y = {}", test.f, b, y);
for p in Sieve::get_primes(test.f as usize) {
if test.is_kth_power( p as u32, y) {
return true;
}
}
false
}
}