"Coprime" Example Sentences
1. The fractions 3/8 and 5/7 are coprime.
2. Two numbers are coprime if their greatest common divisor is 1.
3. The coprime pairs of numbers between 1 and 10 are (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (2,3), (2,5), (2,7), (2,9), (3,4), (3,5), (3,7), (3,8), (4,5), (4,7), (4,9), (5,6), (5,7), (5,8), (5,9), (6,7), (7,8), (7,9), (8,9), and (10,1).
4. The number of coprime pairs of integers between 1 and n is given by Euler's totient function, phi(n).
5. The prime factorization of two coprime numbers has no common factors.
6. The modular multiplicative inverse of a number a modulo m exists if and only if a and m are coprime.
7. The Chinese Remainder Theorem states that if m and n are coprime, then the system of equations x ≡ a (mod m) and x ≡ b (mod n) has a solution.
8. Bezout's Identity states that if a and b are coprime, then there exist integers x and y such that ax + by = 1.
9. The coprime integers a and b are said to be relatively prime.
10. If a and b are coprime, then ab and a + b are also coprime.
11. The product of coprime integers is always coprime with their sum.
12. The coprime integers a and b are said to be mutually prime.
13. The coprime pairs of integers (a,b) that satisfy a^2 + b^2 = c^2 are known as Pythagorean triples.
14. The coprime integers a and b are said to be prime to each other.
15. The coprime integers a and b are said to be free of common factors.
16. The probability that two randomly chosen integers are coprime is 6/π^2.
17. The fraction of positive integers less than n that are coprime to n is given by phi(n)/n.
18. The number of distinct prime factors of two coprime numbers is equal to the sum of the number of distinct prime factors of each number.
19. The Euler product formula states that the sum of the reciprocals of the positive integers that are coprime to n is equal to the product of (1 - 1/p) over all prime factors p of n.
20. The Gaussian integers are numbers of the form a + bi where a and b are coprime integers and i is the imaginary unit.
21. The Eisenstein integers are numbers of the form a + bω where a and b are coprime integers and ω = e^(2πi/3) is a complex cubic root of unity.
22. The Euclidean algorithm can be used to find the greatest common divisor of two coprime integers.
23. The Bertrand postulate states that for every integer n > 1, there exists a prime p such that n < p < 2n. This theorem can be used to show that there always exists a coprime integer to any given integer.
24. The principle of inclusion-exclusion can be used to count the number of positive integers less than n that are not coprime to any of a given set of integers.
25. The Miller-Rabin primality test can be used to determine whether a given number is coprime to a given base or not.
26. The Carmichael function lambda(n) is defined as the smallest integer m such that a^m ≡ 1 (mod n) for all coprime integers a and n. It can be used to improve the efficiency of certain cryptographic algorithms.
27. The Mobius function μ(n) is defined as μ(n) = 1 if n is square-free and has an even number of prime factors, μ(n) = -1 if n is square-free and has an odd number of prime factors, and μ(n) = 0 if n is not square-free. It can be used to compute the sum of the divisors of a given number.
28. The Legendre symbol (a/p) is defined as 1 if a is coprime to p and a is a quadratic residue modulo p, -1 if a is coprime to p and a is a quadratic nonresidue modulo p, and 0 if a is not coprime to p. It can be used to determine whether a given integer is a quadratic residue modulo a given prime or not.
29. The Jacobi symbol (a/n) is a generalization of the Legendre symbol for any odd integer n and any integer a. It can be used to determine whether a given integer is a quadratic residue modulo a given odd integer or not.
30. The Hardy-Littlewood conjecture states that the number of coprime tuples (a_1, a_2, ..., a_k) of positive integers less than x that satisfy a_1 + a_2 + ... + a_k = P(x), where P(x) is some polynomial function, is asymptotic to a certain constant times x^(k-1)/log(x).
Common Phases
1. "The numbers are
coprime."
2. "We need to find
coprime numbers for this problem."
3. "The two integers are
coprime to each other."
4. "
Coprime pairs have no common factors."
5. "Let's check if these numbers are
coprime."
6. "
Coprime fractions cannot be simplified further."
7. "It's important to choose
coprime values for this equation."