Code shared from the Rust Playground

playground.rs

/// Calculates the value of the first identity's right-hand side: n - 1.
///
/// This function directly computes the right-hand side of the stated identity:
/// floor( (sum_{k=n}^inf 1/k^2)^-1 ) = n - 1.
///
/// # Arguments
/// * `n`: A positive integer (u64).
///
/// # Returns
/// The result (n - 1) as a u64.
///
/// # Panics
/// Panics if n is 0, as the identity is stated for a positive integer n.
pub fn identity_k_squared(n: u64) -> u64 {
    if n == 0 {
        panic!("n must be a positive integer.");
    }
    // Since n is a positive integer, n >= 1.
    // The result is n - 1.
    n - 1
}

/// Calculates the value of the second identity's right-hand side: 2 * n * (n - 1).
///
/// This function directly computes the right-hand side of the stated identity:
/// floor( (sum_{k=n}^inf 1/k^3)^-1 ) = 2 * n * (n - 1).
///
/// # Arguments
/// * `n`: A positive integer (u64).
///
/// # Returns
/// The result (2 * n * (n - 1)) as a u64.
///
/// # Panics
/// Panics if n is 0, as the identity is stated for a positive integer n.
pub fn identity_k_cubed(n: u64) -> u64 {
    if n == 0 {
        panic!("n must be a positive integer.");
    }
    // The result is 2 * n * (n - 1).
    // The calculations are safe from overflow for reasonable u64 values of n.
    // For large n, one should consider using u128 or BigInt.
    2 * n * (n - 1)
}

// Example usage and verification (optional, for testing)
fn main() {
    let n1 = 5;
    let res1 = identity_k_squared(n1);
    println!("For n = {}, floor((sum 1/k^2)^-1) = n - 1 = {}", n1, res1); // Output: 4

    let n2 = 10;
    let res2 = identity_k_cubed(n2);
    // 2 * 10 * (10 - 1) = 20 * 9 = 180
    println!("For n = {}, floor((sum 1/k^3)^-1) = 2n(n - 1) = {}", n2, res2); // Output: 180
}

https://play.rust-lang.org/?version=stable&mode=debug&edition=2024&gist=5b89bf8b604e088f98f919290ba48d31