Euler’s Theorem

Euler’s Theorem: Euler’s theorem states that for any integer a that is coprime with a positive integer m, the remainder of a^ϕ(m) divided by m is 1. We focus on proving Euler’s Theorem because Fermat’s Theorem is essentially a specific instance of it. This relationship arises because when p is a prime number, ϕ(p) equals p-1, thus making Fermat’s Theorem a subset of Euler’s Theorem under these conditions.

Euler’s theorem is a fundamental result in number theory, named after the Swiss mathematician Leonhard Euler. It states a relationship between the number theory functions and concepts of modular arithmetic. In this article, we will discuss Euler’s Theorem, including its statement and proof.

What is Euler’s Theorem?

Euler’s Theorem is a fundamental concept in number theory. It states that if n and a are coprime positive integers, meaning that they have no mutual proper dividers other than m=1, then a^ϕ(n) and 1 are relative primes in modulo n.

Euler’s Theorem is a generalization of Fermat’s Little Theorem and serves as a basis for simplifying complex problems into computationally less expensive ones.

Euler’s Theorem Statement

Euler’s Theorem states if a and n are coprime positive integers, then:

a^ϕ(n) ≡ 1 (mod n)

Where,

• ϕ(n) is Euler’s totient function, and
• ≡ denotes equivalence,
• mod n represents congruence modulo n.

Euler’s Totient Function

Formally, for a positive integer n, ϕ(n) is defined as follows:

ϕ(n) = count of integers 1 ≤ a <n such that gcd (a,n)=1

Where:

• gcd(a,n) denotes the greatest common divisor of a and n.
• ϕ(n) represents the totient function of n.

Euler’s Theorem Formula

Statement of Euler’s Theorem can be used as formula for further calculations, i.e.,

a^ϕ(n) ≡ 1 (mod n)

Where,

• a is any integer coprime to n,
• n is a positive integer,
• ϕ(n) is Euler’s totient function, and
• ≡ denotes equivalence,
• mod n represents congruence modulo n.

Example Showing Euler’s Theorem Formula

Problem: Verify Euler’s Theorem for a = 3 and n = 8.

Solution: First, we calculate ϕ(8). The numbers less than 8 that are coprime to 8 are 1, 3, 5, and 7. Thus, ϕ(8)=4.

Next, calculate 3⁴ and find its remainder when divided by 8

3⁴ = 81

Now, find 81 mod 8

81 mod 8 ≡ 1

Thus, 3⁴ ≡ 1 (mod 8), which verifies Euler’s Theorem.

Historical Background of Euler’s Theorem

Euler’s theorem is named after the Swiss mathematician Leonhard Euler. Euler made numerous contributions to various branches of mathematics during the 18th century, and his work laid the groundwork for much of modern mathematics. Euler’s theorem specifically relates to modular arithmetic and the concept of totient function.

The theorem itself is closely related to Euler’s earlier work on Fermat’s Little Theorem. While Fermat’s Little Theorem states a special case of Euler’s theorem, Euler’s theorem provides a more general formulation.

Proof of Euler’s Theorem

Let φ(n) = k, and let {a₁, . . . , a_k} be a reduced residue system mod n.

For some a_i in {a₁, . . . , a_k}

Since (a, n) = 1, {aa₁, . . . , aa_k} is another reduced residue system mod n.

Since this is the same set of numbers mod n as the original system, the two systems must have the same product mod n:

(aa₁)· · ·(aa_k) = a₁ · · · a_k (mod n)

⇒ a^k (a₁ · · · a_k) = a₁ · · · ak (mod n)

Now each a_i is invertible mod n, so multiplying both sides by a₁⁻¹ · · · a_k⁻¹ , We get

a^k = 1 (mod n)

or a^φ(n) = 1 (mod n).

Applications of Euler’s Theorem

Euler’s Theorem has many applications in a wide range of areas, such as mathematics and even elsewhere. Here are some notable applications:

• RSA Encryption

Euler’s theorem is foundational in modern cryptography, particularly in the RSA encryption algorithm. RSA utilizes Euler’s theorem in the process of encryption and decryption. In RSA, the public and private keys are generated in such a way that they are inverses of each other modulo φ(n), where n is the product of two large prime numbers.

• Problem Solving in Number Theory

Euler’s theorem is a powerful tool in solving number theory problems involving divisibility, remainders, and the properties of numbers in different number systems.

• Primality Testing

Euler’s theorem is used in primality testing algorithms, such as the Fermat primality test. While this test is not infallible (it can give false positives for Carmichael numbers), it offers a quick way to check for non-prime numbers. If for some a coprime with φ(n) ≡ 1 (mod n), then n is not prime.

• Mathematical Proofs

Euler’s theorem is a general case for proofs enabling modular arithmetic, divisibility tests and number theory identities and it provides clear and convincing mathematical arguments that are the foundation of rigorous mathematical analysis.

Euler’s Theorem Examples

Example 1: Find the remainder when 3¹⁰⁰ is divided by 7.

Solution:

Since 7 is a prime number, ϕ(7) = 7−1 = 6.

According to Euler’s Theorem,

3⁶ ≡ 1 (mod 7).

Now, 3¹⁰⁰ can be rewritten as 3^6×16+4.

modular exponentiation:

3¹⁰⁰ ≡ (3⁶)¹⁶ × 3⁴ ≡ 1¹⁶ × 81 ≡ 4 (mod 7).

So, when 3¹⁰⁰ is divided by 7, the remainder is 4.

Example 2: Find the remainder when 7²⁰ is divided by 21.

Solution:

Since 21 can be factored into 3 × 7, we have ϕ(21) = (3 – 1) ( 7-1 ) = 2 × 6 = 12.

According to Euler’s Theorem,

7¹² ≡ 1 (mod 21).

Now, 7²⁰ can be expressed as 7^12×1+8.

modular exponentiation:

7²⁰ ≡ (7¹²)¹ × 7⁸ ≡ 1¹ × 5764801 ≡ 1 (mod 21).

So, when 7²⁰ is divided by 21, the remainder is 1.

Practice Questions on Euler’s Theorem

Q1: Find the remainder when 2⁵⁰ is divided by 11.

Q2: Calculate the remainder when 5¹⁰⁰ is divided by 17.

Q3: Determine the remainder when 3⁷⁵ is divided by 13.

Q4: Find the remainder when 4⁴⁰ is divided by 9.

Q5: Find the remainder when 10²⁵ is divided by 8.

FAQs on Euler’s Theorem

What is Euler’s Theorem?

Euler’s Theorem verifies that if a and n are coprime and positive integers, then a^ϕ(n) ≡ 1 (mod n), where ϕ(n) represents the result of Euler’s totient function, i.e. the number of positive integers less than n that are coprime to n.

What is the significance of Euler’s Theorem?

Euler’s Theorem holds importance to number theory and cryptology. That result forms the base for all the other outcomes that provide an understanding for the properties of modular arithmetic and is necessary for various cryptographic systems like RSA encryption system.

Can Euler’s Theorem be applied to composite moduli?

The fact that the Euler’s Theorem can be applied to composite moduli becomes clear, provided that for the condition a and n are coprime positive integers to be satisfied.

How can we use Euler’s Theorem even today?

Euler’s Theorem has been of relevance in number theory and its extension in cryptography with the main application to RSA encryption. Besides, it is also a tool in the number theory, group theory

What are Euler’s Theorem’s general expressions?

Euler’s Theorem, sometimes extended to Carmichael’s Theorem, Fermat’s Little Theorem, and multiple integers, forms the foundations of number theory.