Famous Theorems of Mathematics/Number Theory/Totient Function - Wikibooks, open books for an open world (2024)

FAQs

What are the theorems in elementary number theory? ›

Elementary Number Theory

Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity.

Euler's totient function (also called the Phi function) counts the number of positive integers less than n that are coprime to n.

Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring. ). It is also used for defining the RSA encryption system.

The formula basically says that the value of Φ(n) is equal to n multiplied by-product of (1 – 1/p) for all prime factors p of n. For example value of Φ(6) = 6 * (1-1/2) * (1 – 1/3) = 2.

Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics.

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2.

It states that the remainder of ap−1 when divided by a prime p that doesn't divide a is 1. We then state Euler's theorem which states that the remainder of aϕ(m) when divided by a positive integer m that is relatively prime to a is 1.

THE TAU FUNCTION

The function τ(n) counts how many divisors n has. This count includes 1 and n. (τ is a Greek letter and is called “tau.”) The first few values are τ(1) = 1, τ(2) = 2, τ(3) = 2, τ(4) = 3, τ(5) = 2, τ(6) = 4, τ(7) = 2, τ(8) = 4, τ(9) = 3, and τ(10) = 4.

The Euler's Totient Function counts the numbers lesser than a number say n that do not share any common positive factor other than 1 with n or in other words are co-prime with n. Hence, there are 4 numbers(1,3,5 and 7) that are lesser than 8 and are co-prime with it.

Wilson's theorem states that any prime p divides (p − 1)! + 1, with n! being the factorial notation for 1 × 2 × 3 × 4 × ⋯ × n. Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p – a is an integer multiple of p.

Why is 51 not a prime number? ›

Is 51 a prime number? 51 is not a prime number because it has 3 and 17 as divisors, as well as itself and 1. In other words, 51 has four factors.

Properties of Euler's Totient Function

Φ is the symbol used to denote the function. The function deals with the prime numbers theory. The function is applicable only in the case of positive integers. For ϕ (n), one can find two multiplicative prime numbers to calculate the function.

There are several ways: Limit of ratio between consecutive fibonacci numbers: ϕ=Fn+1Fn as n→∞ Continued square root: φ=√1+√1+√1+⋯ Continued fraction: φ=1+11+11+11+⋯

THE PHI FUNCTION

The first few values of this important function, called the Euler ϕ function, are given by ϕ(1) = 1, ϕ(2) = 1, ϕ(3) = 2, ϕ(4) = 2, ϕ(5) = 4, ϕ(6) = 2 (since only 1 and 5 are relatively prime to 6), ϕ(7) = 6, ϕ(8) = 4, ϕ(9) = 6, and ϕ(10) = 4.

It comes from the Latin tot--"that many, so many" (as in "total"). If totient has a similar origin, than it would mean "that many times" or "all the times". It probably refers to "all the numbers" coprime with n.

Elementary number theory then assures us that there are integers s and k satisfying u = sv + kpⁿ; translating s by a suitable multiple of pⁿ, we may suppose that 0 ≤ s < pⁿ. This implies that r − s = uv⁻¹ − s = kv⁻¹ pⁿ, whence (11) holds.

Theorem 1: The sum of probability of happening and not happening of any given event is always unity, i.e., equals 1. Theorem 2: The probability of an impossible event is always equal to 0. Theorem 3: The sure events always have 1 as a probability. Theorem 4: The probability of any event is always between 0 to 1.

Beginning with any finite collection of primes—say, a, b, c, …, n—Euclid considered the number formed by adding one to their product: N = (abc⋯n) + 1. He then examined the two alternatives: (1) If N is prime, then it is a new prime not among a, b, c, …, n because it is larger than all of these.