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Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 2 and can be written as 3 × 3.
This corresponds to a set of y values whose product is a square number, i.e. one whose factorization has only even exponents. The products of x and y values together form a congruence of squares. This is a classic system of linear equations problem, and can be efficiently solved using Gaussian elimination as soon as the number of rows exceeds ...
Printable version; In other projects ... A perfect square is an element of algebraic structure that is equal to the square of another element. Square number, a ...
For example, to factor =, the first try for a is the square root of 5959 rounded up to the next integer, which is 78. Then b 2 = 78 2 − 5959 = 125 {\displaystyle b^{2}=78^{2}-5959=125} . Since 125 is not a square, a second try is made by increasing the value of a by 1.
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as ...
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. [3]
The first perfect squared square discovered, a compound one of side 4205 and order 55. [1] Each number denotes the side length of its square. Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.)