Number Theory Terminology, Part 1

Number theory terminology relating to chapter 1 of "Elementary Number Theory," by James Strayer.

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Last updated: May 4, 2024

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Hint	Answer
Term for a number that has no positive divisors other than 1 and itself Ex: 2, 3, 5, 7, 11, 13, etc.	Prime
What is the term for a number that has positive divisors other than 1 and itself? Ex: 4, 6, 8, 10, 12, 14, 15, etc.	Composite
The greatest number that divides both of a pair of two numbers	Greatest common divisor
The smallest number that is a multiple of both of a pair of two numbers	Least common multiple
Name of the theorem that states that a number can be written uniquely as a product of powers of primes, known as prime factorization?	Fundamental Theorem of Arithmetic
Name of the theorem that states that there are infinitely many primes of the form a + nb, where n is an integer and GCD(a, b) = 1.	Dirichlet's Theorem
Name of the theorem that states that the number of primes less than or equal to a number X is asymptotically approaches x/ln(x)?	Prime Number Theorem
Term for when the k-th power of a prime number, p, is the highest power of p that divides a number b, denoted by (p^k) \|\| b?	Exactly divides
If we have an integer n, where all primes appearing in the prime factorization of n have an exponent of at least 2, what adjective do we give to n?	Powerful
Given any integers a, b, we can find unique integers q, r such that 0 <= r < b, and a = bq + r. What is the name of this theorem?	Division Algorithm
What is the algorithm we use to find the greatest common factor of two numbers?	Euclidean Algorithm
What is the term for methods that eliminate composite numbers from a list of integers, leaving only prime numbers remaining?	Sieve
What Greek mathematician pioneered the above methods?	Eratosthenes
What is the term for primes of the form (2^n) - 1, where n is a positive integer? Examples: 3, 7, 31, etc.	Mersenne primes
What is the term for primes of the form (2^(2^n)) + 1, where n is a nonnegative integer? Examples: 3, 5, 17, etc.	Fermat primes
What is the name of the conjecture that every even integer > 2 can be written as the sum of two (not necessarily distinct) prime numbers?	Goldbach conjecture