In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization...

by the square root of the unique square-free integer D that satisfies b2 − 4c = De2 for some integer e. If D is positive, the quadratic integer is real...

{n}}} of a nonnegative integer n is an algebraic integer, but is irrational unless n is a perfect square. If d is a square-free integer then the extension...

mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself...

is given by (1, ζ, ζ 2, ..., ζ p−2). If d {\displaystyle d} is a square-free integer and K = Q ( d ) {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)} is...

{\displaystyle {a+b{\sqrt {c}} \over d},} for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic...

({\sqrt {d}})} where d {\displaystyle d} is a (uniquely defined) square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d...

A square-free integer is an integer that is not divisible by any square other than 1. square number A square number is an integer that is the square of...

sum of 427 when excluding a square of side length 7 {\displaystyle 7} ; this sum represents the largest square-free integer over a quadratic field of class...

semiprime. 58 is not divisible by any square number other than 1, making it a square-free integer A semiprime that is not square numbers is called a squarefree...

Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial Rational number...

d be a square-free integer. The field Q(√d) is a quadratic extension of Q. The class number of Q(√d) is one if and only if the ring of integers of Q(√d)...

This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to...

and 71. 69 is not divisible by any square number other than 1, making it a square-free integer. 69 is a Blum integer since the two factors of 69 are both...

would make it divisible by a square number). Common examples of square-free elements include square-free integers and square-free polynomials. Prime number...

group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent...

construction chooses a square-free integer b {\displaystyle b} as the radix of the base- b {\displaystyle b} notation for the integers, such that there exists...

Every n in Mx can be written as n = m2r with positive integers m and r, where r is square-free. Since only the k primes p1, ..., pk can show up (with...

decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater...

positive integer n is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} holds for every integer a coprime...

F=\mathbb {Q} ({\sqrt {m}})} , where m {\displaystyle m} is a nonzero square-free integer (we can include the case Q ( 1 ) = Q {\displaystyle \mathbb {Q} ({\sqrt...

countably infinite set of ideal classes. If d {\displaystyle d} is a square-free integer (a product of distinct primes) other than 1, then Q ( d ) {\displaystyle...

\mathbb {Q} ({\sqrt {a}})/\mathbb {Q} } where a {\displaystyle a} is a square-free integer. Then, the multiplication map by a {\displaystyle {\sqrt {a}}} on...

and is square-free, meaning it is not divisible by the square of any prime number. Case 2: D {\textstyle D} is equal to four times an integer m {\textstyle...

geometric sum formula then shows that G(a, b, 2m) = 0. If c is an odd square-free integer and gcd(a, c) = 1, then G ( a , 0 , c ) = ∑ n = 0 c − 1 ( n c ) e...

speed offered by the fast inverse square root trick came from treating the 32-bit floating-point word as an integer, then subtracting it from a "magic"...

radical of any integer n {\displaystyle n} is the largest square-free divisor of n {\displaystyle n} and so also described as the square-free kernel of n...

numbers as the non-negative integers 0, 1, 2, 3, ..., while others start with 1, defining them as the positive integers 1, 2, 3, ... . Some authors acknowledge...

using a computer's word size is the most common choice. If m were a square-free integer, this would only allow a ≡ 1 (mod m), which makes a very poor PRNG;...

multiple consequences, including the following two: Let n be an odd square-free integer. Assuming the Birch and Swinnerton-Dyer conjecture, n is the area...