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...

{\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...

{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...

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

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...

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...

number if S = {1}, A square if gcd(S) is even, A cube if gcd(S) is divisible by 3, A square-free integer if max(S) = 1, A cube-free integer if max(S) ≤ 2, A...

of integers, the semiprime ideals are the zero ideal, along with those ideals of the form n Z {\displaystyle n\mathbb {Z} } where n is a square-free integer...

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)...

partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only...

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...

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...

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...

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...

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...

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...

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...

solutions of a few fairly simple Diophantine equations. For a given square-free integer n, define A n = # { ( x , y , z ) ∈ Z 3 ∣ n = 2 x 2 + y 2 + 32 z...

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...

\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...

) , {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,),} where d is a square-free integer, and that the narrow class group of K is trivial. Suppose that {...

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...

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...

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...

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"...

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;...