An integer that is not divisible by the square of any prime; equivalently, in its prime factorization every prime appears with exponent 0 or 1.

Principle

Squarefree integers are those whose prime-power factorization has no exponent ≥2; the radical of n (product of distinct prime factors) equals |n| for squarefree n, and multiplicative functions often simplify on this subclass.

Demonstration

Examples: 30 = 2×3×5 is squarefree, 18 = 2×3^2 is not squarefree because 3^2 divides 18, and 1 is conventionally squarefree because it has no prime divisors.

Misapplication

Confusing 'squarefree' with 'not a perfect square' (e.g., 6 is squarefree but not a square) or overlooking sign and zero (0 is divisible by every square and thus not squarefree) are common mistakes.

Consequence

Squarefree integers play key roles in Möbius and Liouville function behavior, in counting problems (squarefree density ~6/π^2), and in structural results where restricting to squarefree inputs simplifies multiplicative formulas or local-global arguments.

Reversal

The opposite class consists of integers divisible by a prime square (squareful or not squarefree); powerful numbers are an extreme reversal where every prime dividing n does so with exponent at least 2.

Boundary

Definition applies to integers; 0 is not squarefree, 1 is squarefree by convention, and in algebraic number fields the analogous notion requires considering prime ideal squares—care is needed when generalizing from Z to rings of integers in number fields.

Semantic Tension

Tension arises between the elementary integer notion of squarefree and related algebraic notions such as squarefreeness of polynomials or squarefree ideals in rings—similar in spirit but requiring different local criteria and techniques.

Synthesis

A squarefree integer is one whose prime factorization contains no repeated primes; this simple combinatorial condition has far-reaching consequences in multiplicative number theory, density estimates, and simplifications of arithmetic functions.