Understanding Euler's Totient Function and Finding the Smallest Integer

The Euler's Totient Function, denoted by φ(n), counts the number of positive integers up to n that are coprime to n. That is, for a number n, φ(n) counts how many integers in the range 1 to n are relatively prime to n. The totient function is fundamental in number theory and is often used in various cryptographic and algorithmic applications.

Definition and Formula

Euler's Totient Function is formally defined as follows:

Definition

The function φ(n) counts the number of integers up to n that are coprime to n. Two integers are coprime if their greatest common divisor (GCD) is 1.

The mathematical formula for the totient function is given by:

Formula

For a number n with the prime factorization n p_1^{k_1} p_2^{k_2} cdots p_m^{k_m}, the totient function is:

φ(n) n left(1 - frac{1}{p_1}right)left(1 - frac{1}{p_2}right) cdots left(1 - frac{1}{p_m}right)

Steps to Find the Smallest Integer n such that φ(n) k

To find the smallest integer n such that φ(n) k, follow these steps:

Identify Primes

Start with the smallest primes: 2, 3, 5, 7, etc., and consider their combinations.

Calculate n for Different Combinations

Use the formula for φ(n) to solve for n given k:

n frac{k}{prod_{i1}^{m} left(1 - frac{1}{p_i}right)}

Choose combinations of primes p_1 p_2 cdots p_m and calculate n iteratively.

Check for Integer Values

Ensure that the calculated n is an integer.

Find the Smallest n

Keep track of the smallest n found that satisfies φ(n) k.

Example

Let's find the smallest integer n such that φ(n) 8:

1. Start with combinations of primes:

2. For n 2^3 8:

φ(8) 8 ? (1 - 1/2) 4

This is not valid since φ(8) ≠ 8.

3. For n 3^2 9:

φ(9) 9 ? (1 - 1/3) 6

This is not valid since φ(9) ≠ 8.

4. For n 2 ? 5 10:

φ(10) 10 ? (1 - 1/2)(1 - 1/5) 4

This is not valid since φ(10) ≠ 8.

5. For n 2^3 ? 3 24:

φ(24) 24 ? (1 - 1/2)(1 - 1/3) 8

This is valid since φ(24) 8.

6. Check smaller combinations:

For n 15:

φ(15) 15 ? (1 - 1/3)(1 - 1/5) 8

This is also valid since φ(15) 8.

Conclusion: The smallest n such that φ(n) 8 is n 15.

Final Note

This process may involve trial and error with various combinations of primes. However, using smaller primes first usually leads to finding the smallest n efficiently. Additionally, Euler's Totient Function can be calculated as a limit of the Riemann Zeta Function in certain contexts.

In the context of algorithmic challenges, the problem may be approached differently. For example, with a limit of 1.5GB memory, a direct sieving method was used to optimise time at the cost of memory. The sieve built factorizations and precomputed φ values, allowing for O(1) lookup times for inverse φ values.

Overall, mastering the steps and understanding the properties of the Euler's Totient Function will significantly aid in solving problems related to number theory and related areas.