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Questions and Answers
What is the value of φ(p) for prime p?
What is the value of φ(p) for prime p?
If gcd(a, b) = 1, what is the value of φ(ab)?
If gcd(a, b) = 1, what is the value of φ(ab)?
What is the Euler's totient function, φ(n), used in?
What is the Euler's totient function, φ(n), used in?
What is the value of φ(p^k) for prime p and k ≥ 1?
What is the value of φ(p^k) for prime p and k ≥ 1?
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What is the definition of φ(n), the Euler's totient function?
What is the definition of φ(n), the Euler's totient function?
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How is the greatest common divisor (GCD) calculated in the calculateGCD function?
How is the greatest common divisor (GCD) calculated in the calculateGCD function?
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If the resulting GCD in the calculateGCD function is equal to 1, x and i are not relatively prime.
If the resulting GCD in the calculateGCD function is equal to 1, x and i are not relatively prime.
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A set of numbers are said to be ____ when their greatest common divisor is 1
A set of numbers are said to be ____ when their greatest common divisor is 1
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Define what a divisor is according to number theory
Define what a divisor is according to number theory
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What is the greatest common divisor (GCD) of integers 35 and 21?
What is the greatest common divisor (GCD) of integers 35 and 21?
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A composite number is a positive integer with no divisors other than itself and 1. (True/False)
A composite number is a positive integer with no divisors other than itself and 1. (True/False)
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Explain what a reduced residue class mod m is.
Explain what a reduced residue class mod m is.
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Study Notes
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.
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Description
Explore Euler's influential work in number theory, including his totient function and theorem. Learn about the properties and applications of these fundamental concepts.
