"Two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1."
Two numbers that do not have any common divisors other than 1.
Divisibility: A fundamental concept in number theory that involves whether one integer can be divided exactly by another integer.
GCD (Greatest Common Divisor): The largest positive integer that divides two or more numbers without leaving any remainder.
Euclid's algorithm: A mathematical procedure used to find the GCD of two numbers.
Prime numbers: Integers that are only divisible by 1 and themselves.
Fundamental theorem of arithmetic: Every integer greater than 1 can be expressed as a product of prime numbers, and this expression is unique except for the order of the factors.
Euler's totient function: A function that returns the number of positive integers that are less than or equal to a given integer and are relatively prime to it.
RSA encryption: A cryptographic algorithm based on the difficulty of factoring the product of two large prime numbers.
Chinese remainder theorem: A mathematical theorem that describes the relationship between systems of linear equations and their solutions.
Bezout's identity: A theorem that states that for any two integers a and b, there exist integers x and y such that ax + by = gcd(a,b).
Modular arithmetic: A type of arithmetic that deals with the remainder of integers after being divided by a fixed integer, called the modulus.
Carmichael function: A function that returns the smallest positive integer that satisfies a certain congruence condition.
Fermat's little theorem: A theorem that states that if p is a prime number and a is any integer, then a^p-1 is congruent to 1 mod p.
Extended Euclidean algorithm: A variation of Euclid's algorithm that finds the GCD of two integers and also determines the coefficients of Bezout's identity.
Euler's theorem: A generalization of Fermat's little theorem that applies to any integer, not just primes.
Pythagorean triples: Sets of three integers that satisfy the Pythagorean theorem, a^2 + b^2 = c^2, where a and b are the two shorter sides of a right triangle, and c is the hypotenuse.
Primitive coprimes: Two positive integers a and b are primitive coprimes if their greatest common divisor (GCD) is 1.
Strong coprimes: Two positive integers a and b are strong coprimes if their product ab is a square-free integer.
Square coprimes: Two positive integers a and b are square coprimes if their sum and difference are not perfect squares.
Twin coprimes: Two positive integers a and a+2 (or a-2) are twin coprimes if they are both prime.
A-range coprimes: Two positive integers a and b are A-range coprimes if they are not divisible by any prime less than or equal to a.
Beal coprimes: Three positive integers a, b, and c are Beal coprimes if ax + by = cz has no nontrivial solutions with x, y, and z coprime to each other.
Fermat coprimes: Two positive integers a and b are Fermat coprimes if their product ab can be expressed as the sum of two squares in only one way.
Lehmer coprimes: Two positive integers a and b are Lehmer coprimes if a/b and b/a are both irreducible fractions.
Lucas coprimes: Two positive integers a and b are Lucas coprimes if their linear combination an + bm is never a perfect square when n and m are nonnegative integers.
Wilf coprimes: Two positive integers a and b are Wilf coprimes if they have no common factor larger than 1 and satisfy the equation a^2 + b^2 = c^2 where c is also an integer.
"This is equivalent to their greatest common divisor (GCD) being 1."
"Any prime number that divides a does not divide b, and vice versa."
"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."
"Two integers a and b are coprime if the only positive integer that divides both of them is 1. This property has various applications in number theory and mathematical proofs."
"No, the only positive integer that is a divisor of both coprime numbers is 1."
"The concept of coprime numbers is closely related to prime factorization, as prime numbers are divisors that can affect coprimality."
"No, by definition, the only positive integer that can divide both coprime numbers is 1."
"No, if two numbers are coprime, it means that any prime number dividing one of them does not divide the other."
"Yes, two prime numbers are always coprime because they do not have any common divisors other than 1."
"No, the coprimality of two numbers does not depend on their magnitude, but on their shared divisors."
"No, a number cannot be coprime with itself since it would have at least one non-trivial common divisor."
"Coprimality refers to the absence of common divisors greater than 1 between two numbers, while primality refers to the number itself being divisible only by 1 and itself."
"Yes, coprimality is not affected by the sign of the numbers. The concept applies to both positive and negative integers."
"The property of having a greatest common divisor of 1 can be used to determine if two numbers are coprime or not."
"No, coprimality is determined based on the absence of common divisors greater than 1, rather than following specific rules or formulas."
"Yes, being coprime and being relatively prime refer to the same concept. The terms are used interchangeably in number theory."
"No, coprime numbers do not have any common factors other than 1. That is what makes them coprime in the first place."