Kvadrattal

Kvadrattal, or square numbers, are integers that are the square of another integer. Formally, a kvadrattal is any integer of the form n^2 with n ∈ Z. Depending on the convention for natural numbers, 0 may or may not be included; since 0^2 = 0, it is a square number in many contexts.

Examples of kvadrattal include 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

Key properties of kvadrattal include parity and modular behavior. If n is even, then n^2 is divisible by 4; if n is odd, then n^2 ≡ 1 mod 4. Therefore every kvadrattal is congruent to 0 or 1 modulo 4. In prime factorization, kvadrattal have even exponents for all primes; equivalently, a number is a kvadrattal if and only if its prime factors appear with even exponents, meaning its square root is an integer.

Arithmetic and geometry offer simple interpretations. The difference between consecutive kvadrattal is an odd number: (n+1)^2 − n^2 = 2n+1. The sum of the first n odd numbers equals n^2. A kvadrattal represents the area of a square with integer side length, linking them to lattice geometry and combinatorics.

Patterns in base 10 include a restricted set of final digits: kvadrattal can end only in 0, 1, 4, 5, 6, or 9; they never end in 2, 3, 7, or 8. They appear in various number theory contexts, including Pythagorean triples and modular arithmetic.

See also: Perfect square, Square root, Pythagorean theorem, Lattice point.