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...

{\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...

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...

({\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...

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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

\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...

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;...

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...

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...

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...

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...

Sloane, N. J. A. (ed.). "Sequence A001110 (Square triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A...

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...

A122400 (Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation...

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"...