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The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition.
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). [ 4 ] [ 5 ] This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } ). [ 6 ]
Then () = means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); () = means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20).
Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive divisors. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition. [7]
The two first subsections, are proofs of the generalized version of Euclid's lemma, namely that: if n divides ab and is coprime with a then it divides b. The original Euclid's lemma follows immediately, since, if n is prime then it divides a or does not divide a in which case it is coprime with a so per the generalized version it divides b.
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Numbers of the form + (+,) with a, b any coprime integers, a > b > 0, are called generalized Fermat numbers. An odd prime p is a generalized Fermat number if and only if p is congruent to 1 (mod 4) .
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.