diff --git a/DIRECTORY.md b/DIRECTORY.md
index 746db33f9b8..bd3c98b36f8 100644
--- a/DIRECTORY.md
+++ b/DIRECTORY.md
@@ -67,13 +67,16 @@
 
   * [Lib](https://github.com/TheAlgorithms/Rust/blob/master/src/lib.rs)
   * Math
+    * [Amicable Numbers Below N](https://github.com/TheAlgorithms/Rust/blob/master/src/math/amicable_numbers.rs)
     * [Baby-Step Giant-Step Algorithm](https://github.com/TheAlgorithms/Rust/blob/master/src/math/baby_step_giant_step.rs)
     * [Extended Euclidean Algorithm](https://github.com/TheAlgorithms/Rust/blob/master/src/math/extended_euclidean_algorithm.rs)
+    * [Fast Inverse Square Root 'Quake' Algorithm](https://github.com/TheAlgorithms/Rust/blob/master/src/math/square_root.rs)
     * [Greatest Common Divisor](https://github.com/TheAlgorithms/Rust/blob/master/src/math/greatest_common_divisor.rs)
     * [Pascal Triangle](https://github.com/TheAlgorithms/Rust/blob/master/src/math/pascal_triangle.rs)
     * [Perfect Numbers](https://github.com/TheAlgorithms/Rust/blob/master/src/math/perfect_numbers.rs)
     * [Prime Check](https://github.com/TheAlgorithms/Rust/blob/master/src/math/prime_check.rs)
     * [Prime Numbers](https://github.com/TheAlgorithms/Rust/blob/master/src/math/prime_numbers.rs)
+    * [Square root with Newton's method](https://github.com/TheAlgorithms/Rust/blob/master/src/math/square_root.rs)
     * [Trial Division](https://github.com/TheAlgorithms/Rust/blob/master/src/math/trial_division.rs)
     * [Miller Rabin](https://github.com/TheAlgorithms/Rust/blob/master/src/math/miller_rabin.rs)
     * [Linear Sieve](https://github.com/TheAlgorithms/Rust/blob/master/src/math/linear_sieve.rs)
diff --git a/README.md b/README.md
index 29bb483c7ca..7c0e1d6f50a 100644
--- a/README.md
+++ b/README.md
@@ -50,8 +50,10 @@ These are for demonstration purposes only.
 
 ## [Math](./src/math)
 
+- [x] [Amicable numbers below N](./src/math/amicable_numbers.rs)
 - [x] [Baby-Step Giant-Step Algorithm](./src/math/baby_step_giant_step.rs)
 - [x] [Extended euclidean algorithm](./src/math/extended_euclidean_algorithm.rs)
+- [x] [Fast Inverse Square Root 'Quake' Algorithm](./src/math/square_root.rs)
 - [x] [Gaussian Elimination](./src/math/gaussian_elimination.rs)
 - [x] [Greatest common divisor](./src/math/greatest_common_divisor.rs)
 - [x] [Greatest common divisor of n numbers](./src/math/gcd_of_n_numbers.rs)
diff --git a/src/math/amicable_numbers.rs b/src/math/amicable_numbers.rs
new file mode 100644
index 00000000000..4d466cd4134
--- /dev/null
+++ b/src/math/amicable_numbers.rs
@@ -0,0 +1,69 @@
+// Operations based around amicable numbers
+// Suports u32 but should be interchangable with other types
+// Wikipedia reference: https://en.wikipedia.org/wiki/Amicable_numbers
+
+// Returns vec of amicable pairs below N
+// N must be positive
+pub fn amicable_pairs_under_n(n: u32) -> Option<Vec<(u32, u32)>> {
+    let mut factor_sums = vec![0; n as usize];
+
+    // Make a list of the sum of the factors of each number below N
+    for i in 1..n {
+        for j in (i * 2..n).step_by(i as usize) {
+            factor_sums[j as usize] += i;
+        }
+    }
+
+    // Default value of (0, 0) if no pairs are found
+    let mut out = vec![(0, 0)];
+    // Check if numbers are amicable then append
+    for (i, x) in factor_sums.iter().enumerate() {
+        if (*x != i as u32) && (*x < n) && (factor_sums[*x as usize] == i as u32) && (*x > i as u32)
+        {
+            out.push((i as u32, *x));
+        }
+    }
+
+    // Check if anything was added to the vec, if so remove the (0, 0) and return
+    if out.len() == 1 {
+        None
+    } else {
+        out.remove(0);
+        Some(out)
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    pub fn test_amicable_numbers_below_n() {
+        // First 10 amicable numbers, sorted (low, high)
+        let expected_result = vec![
+            (220, 284),
+            (1184, 1210),
+            (2620, 2924),
+            (5020, 5564),
+            (6232, 6368),
+            (10744, 10856),
+            (12285, 14595),
+            (17296, 18416),
+            (63020, 76084),
+            (66928, 66992),
+        ];
+
+        // Generate pairs under 100,000
+        let mut result = amicable_pairs_under_n(100_000).unwrap();
+
+        // There should be 13 pairs under 100,000
+        assert_eq!(result.len(), 13);
+
+        // Check the first 10 against known values
+        result = result[..10].to_vec();
+        assert_eq!(result, expected_result);
+
+        // N that does not have any amicable pairs below it, the result should be None
+        assert_eq!(amicable_pairs_under_n(100), None);
+    }
+}
diff --git a/src/math/mod.rs b/src/math/mod.rs
index 821034673e1..efcf841cf14 100644
--- a/src/math/mod.rs
+++ b/src/math/mod.rs
@@ -1,3 +1,4 @@
+mod amicable_numbers;
 mod armstrong_number;
 mod baby_step_giant_step;
 mod extended_euclidean_algorithm;
@@ -29,6 +30,7 @@ mod square_root;
 mod trial_division;
 mod zellers_congruence_algorithm;
 
+pub use self::amicable_numbers::amicable_pairs_under_n;
 pub use self::armstrong_number::is_armstrong_number;
 pub use self::baby_step_giant_step::baby_step_giant_step;
 pub use self::extended_euclidean_algorithm::extended_euclidean_algorithm;
@@ -63,6 +65,6 @@ pub use self::quadratic_residue::cipolla;
 pub use self::random::PCG32;
 pub use self::sieve_of_eratosthenes::sieve_of_eratosthenes;
 pub use self::simpson_integration::simpson_integration;
-pub use self::square_root::square_root;
+pub use self::square_root::{fast_inv_sqrt, square_root};
 pub use self::trial_division::trial_division;
 pub use self::zellers_congruence_algorithm::zellers_congruence_algorithm;
diff --git a/src/math/square_root.rs b/src/math/square_root.rs
index 908f8d55987..42d87e395ef 100644
--- a/src/math/square_root.rs
+++ b/src/math/square_root.rs
@@ -14,12 +14,43 @@ pub fn square_root(num: f64) -> f64 {
     root
 }
 
+// fast_inv_sqrt returns an approximation of the inverse square root
+// This algorithm was fist used in Quake and has been reimplimented in a few other languages
+// This crate impliments it more throughly: https://docs.rs/quake-inverse-sqrt/latest/quake_inverse_sqrt/
+pub fn fast_inv_sqrt(num: f32) -> f32 {
+    // If you are confident in your input this can be removed for speed
+    if num < 0.0f32 {
+        return f32::NAN;
+    }
+
+    let i = num.to_bits();
+    let i = 0x5f3759df - (i >> 1);
+    let y = f32::from_bits(i);
+
+    println!("num: {:?}, out: {:?}", num, y * (1.5 - 0.5 * num * y * y));
+    // First iteration of newton approximation
+    y * (1.5 - 0.5 * num * y * y)
+    // The above can be repeated again for more precision
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
 
     #[test]
-    fn test() {
+    fn test_fast_inv_sqrt() {
+        // Negatives don't have square roots:
+        assert!(fast_inv_sqrt(-1.0f32).is_nan());
+
+        // Test a few cases, expect less than 1% error:
+        let test_pairs = [(4.0, 0.5), (16.0, 0.25), (25.0, 0.2)];
+        for pair in test_pairs {
+            assert!((fast_inv_sqrt(pair.0) - pair.1).abs() <= (0.01 * pair.0));
+        }
+    }
+
+    #[test]
+    fn test_sqare_root() {
         assert!((square_root(4.0_f64) - 2.0_f64).abs() <= 1e-10_f64);
         assert!(square_root(-4.0_f64).is_nan());
     }