kotlin coding test course, implementing Euler's phi function

1. Introduction

There are various algorithm problems to prepare for coding tests. Among them, many problems require mathematical thinking. Today, we will learn about Euler’s Totient Function. The Euler’s Totient Function returns the count of integers that are coprime to n among the integers from 1 to n. This function plays a very important role in number theory and is frequently used in cryptographic algorithms.

2. Understanding Euler’s Totient Function

The Euler’s Totient Function φ(n) has the following properties:

φ(1) = 1
When p is a prime number, φ(p) = p – 1
When p is a prime and k is a natural number, φ(p^k) = p^k – p^(k-1)
If two numbers are coprime, φ(m * n) = φ(m) * φ(n)

The most common method to calculate Euler’s Totient Function is through prime factorization. You can obtain the value through the prime factorization of the given integer n. For example, if n = 12, the prime factorization can be expressed as 2^2 * 3^1, and to find the value of φ(12), the following formula can be used.

φ(n) = n * (1 – 1/p₁) * (1 – 1/p₂) * … * (1 – 1/pₖ)

Here, p₁, p₂, …, pₖ are the prime factors of n. Therefore, for 12, φ(12) is calculated as follows:

φ(12) = 12 * (1 – 1/2) * (1 – 1/3) = 12 * 1/2 * 2/3 = 4

3. Problem Definition

This problem involves implementing a Kotlin function that returns the value of φ(n) for a given integer n. This problem requires not only calculating the value of φ(n) but also considering the efficiency of the algorithm.

**Problem**: Given a natural number n, implement a function in Kotlin that returns the value of Euler’s Totient Function φ(n).

4. Problem Solving Process

4.1 Algorithm Design

An efficient way to calculate Euler’s Totient Function is through prime factorization. The algorithm consists of the following steps:

Set the initial value of result to n for the given natural number n.
For each integer p from 2 to the square root of n:
If p is a prime factor of n, set result = result * (1 – 1/p) and divide n by p.
If n is greater than 1 at the end, treat it as a prime and calculate result = result * (1 – 1/n).
Return result.

4.2 Coding

    
    fun eulerTotient(n: Int): Int {
        var result = n
        var p: Int = 2
        var tempN = n

        while (p * p <= tempN) {
            if (tempN % p == 0) {
                // If p is a prime factor
                while (tempN % p == 0) {
                    tempN /= p
                }
                result *= (p - 1)
                result /= p
            }
            p++
        }

        // Remaining prime
        if (tempN > 1) {
            result *= (tempN - 1)
            result /= tempN
        }

        return result
    }

4.3 Code Explanation

The above code implements Euler’s Totient Function through the following process:

Initialize the variable result to n to store the value of φ(n).
p starts from 2 and iterates over all integers up to the square root of n.
If n is divisible by p, then p is a prime factor of n.
Perform complete division of n by p and update the result.
After processing all prime factors, if n is greater than 1, reflect n as the only prime in the result.

5. Performance Analysis

The above algorithm has a time complexity of O(√n) and is therefore very efficient. It can quickly compute φ(n) even with large amounts of data. Performance is not an issue even for large numbers like 1,000,000.

6. Examples

Here are some examples of using the function:

    
    fun main() {
        println("φ(1) = " + eulerTotient(1)) // 1
        println("φ(12) = " + eulerTotient(12)) // 4
        println("φ(13) = " + eulerTotient(13)) // 12
        println("φ(30) = " + eulerTotient(30)) // 8
        println("φ(100) = " + eulerTotient(100)) // 40
    }

7. Conclusion

Today, we learned about Euler’s Totient Function and how to implement it in Kotlin. This problem is often encountered in coding tests and requires both mathematical thinking and algorithmic approaches, so practice is necessary. Validate the implemented function through various test cases and try tackling extended problems as well.

Through problem-solving tutorials like this, I hope to share knowledge with more people and grow together.

8. References

Concrete Mathematics: A Foundation for Computer Science by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik
Introduction to Algorithms by Thomas H. Cormen et al.
Number Theory: A Modern Introduction by Andrew P. Erdos

kotlin coding test course, implementing Euler’s phi function