KnowraPerfect squarePerfect squareA perfect square is an integer of the form n² for some integer n. Its nonnegative square root is an integer.BriefConnectInteger: A whole number, positive, negative, or zero. Perfect squares are defined within the integers, including zero.Difference of squares: The factorization identity a² − b² = (a − b)(a + b). Differences between consecutive squares factor into consecutive odd integers.Square number: A number obtained by multiplying an integer by itself. This name emphasizes the geometric interpretation of each positive perfect square.Perfect cube: An integer equal to the cube of another integer. Cubes require prime exponents divisible by three, rather than even exponents.Square root: A number whose square equals a given number; the principal square root is nonnegative. An integer is a perfect square exactly when its principal square root is an integer.Divisibility rule: A criterion for deciding whether one integer divides another without performing full division. Remainder patterns provide quick tests for whether an integer could be a square.Figurate number: A number represented by a regular geometric arrangement of points. Perfect squares are the figurate numbers that form square arrays of points.Semisquare: A positive integer that is the product of two primes, with repetition allowed. Its prime-factor pattern need not meet the even-exponent condition for a square.Prime factorization: The expression of an integer greater than one as a product of primes. A positive integer is a square precisely when every prime has an even exponent in its factorization.Modular arithmetic: Arithmetic in which integers are identified by their remainders after division by a fixed modulus. Squares occupy only certain remainder classes, ruling out many candidates.Show all 21Linked from 8 pagesDiophantusRelated: Square numbers and square expressions recur in the equation problems attributed to Diophantus.Lagrange's four-square theoremNarrower topic: Each summand in the theorem is a perfect square, including zero.Quadratic residueRelated: A quadratic residue is a modular counterpart of an ordinary perfect square.Pell equationRelated: The parameter D must not be a perfect square for this equation to have its Pell-type structure.Ramanujan–Nagell equationRelated: Each solution requires 2ⁿ − 7 to be a perfect square.Square root of 2Related: Two is not a perfect square, explaining why its square root is not an integer.Legendre's three-square theoremRelated: Each term in a three-square representation is a perfect square, including zero.Squared triangular numberRelated: Every term is a perfect square, with a triangular number as its square root.