Euler's Contributions to Number Theory

Contributions to Number Theory

• Euler's work in number theory is vast and influential, with many concepts and theorems bearing his name.
• He introduced the concept of the Euler's totient function, φ(n), which counts the number of positive integers up to n that are relatively prime to n.
• Euler's theorem states that if a and n are coprime, then a^(φ(n)) ≡ 1 (mod n).

Euler's Totient Function

• The Euler's totient function, φ(n), is a multiplicative function that counts the number of positive integers up to n that are relatively prime to n.
• φ(n) is defined as the number of k, 1 ≤ k ≤ n, such that gcd(k, n) = 1.
• The totient function is used in many areas of number theory, including cryptography, primality testing, and Diophantine equations.
• Euler's totient function has several important properties, including:
  • φ(p) = p - 1 for prime p
  • φ(p^k) = p^k - p^(k-1) for prime p and k ≥ 1
  • φ(ab) = φ(a)φ(b) if gcd(a, b) = 1

Euler's Contributions to Number Theory

• Euler's work in number theory is vast and influential, with many concepts and theorems bearing his name.

Euler's Totient Function

• The Euler's totient function, φ(n), counts the number of positive integers up to n that are relatively prime to n.
• φ(n) is defined as the number of k, 1 ≤ k ≤ n, such that gcd(k, n) = 1.
• The totient function is used in many areas of number theory, including:
  • Cryptography
  • Primality testing
  • Diophantine equations
• φ(n) has several important properties, including:
  • φ(p) = p - 1, where p is prime
  • φ(p^k) = p^k - p^(k-1), where p is prime and k ≥ 1
  • φ(ab) = φ(a)φ(b), where gcd(a, b) = 1

Euler's Theorem

• Euler's theorem states that if a and n are coprime, then a^(φ(n)) ≡ 1 (mod n).

Introduction to Euler's Totient Function

• Number theory is the study of relationships and properties of numbers.
• Euler's totient function, also known as the phi-function, is a fundamental concept in number theory.
• It is denoted by φ(n) and represents the number of positive integers less than n that are coprime to n.

Background

• Divisibility: an integer b is divisible by an integer a if a leaves no remainder when dividing b.
• Greatest Common Divisor (GCD): the largest divisor shared by two or more integers.
• Coprime: a set of numbers are coprime if their GCD is 1.
• Surjective function: a function f: X → Y is surjective if for every element y in Y, there exists an element x in X such that f(x) = y.

Fundamental Concepts

• Prime numbers: positive integers with no divisors other than themselves and 1.
• Composite numbers: positive integers that are not prime.
• Congruence: an integer a is congruent to an integer b modulo m if a - b is divisible by m.
• Reduced residue class: a set of numbers that are coprime to the modulus m and are not congruent to each other modulo m.

Patterns of the Totient Function

• φ(p) = p - 1 for a prime number p.
• φ(pq) = φ(p)φ(q) for distinct primes p and q.
• φ(n) is a multiplicative function: φ(mn) = φ(m)φ(n) for coprime m and n.
• φ(pr) = pr - pr-1 for a prime number p.

The Surjectivity of the Totient Function

• φ(x) is never odd for integers x ≥ 3.
• φ(x) is only odd when x = 2.
• The codomain of the totient function does not contain every even number.
• There exists a lower bound for φ(x) where no value can φ(x) ≥ x2/p.

Examples and Applications

• φ(6) = 2, as the integers less than 6 that are coprime to 6 are 1 and 5.
• φ(30) = 8, and we can find 4 distinct x1, x2, x3, x4 such that φ(30) = φ(x1) = ... = φ(x4).
• The totient function can be used to find the number of reduced residue classes modulo n.