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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.
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [ 4 ] A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the n {\displaystyle n} th triangular number n ( n + 1 ) 2 {\displaystyle {\tfrac {n(n+1)}{2}}} is square, then ...
Hence all centered square numbers and their divisors end with digit 1 or 5 in base 6, 8, and 12. Every centered square number except 1 is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean triples where the two longest sides differ by 1. (Example: 5 2 + 12 2 = 13 2.)
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 ...
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 ...
To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to the end of each row (n elements), one to the end of each column (n elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above ...
In the above example, the number '13' represents a winning or 'hot' position, whilst the number '2' represents a losing or 'cold' position. Given an integer list with each integer labeled 'hot' or 'cold', the strategy of the game is simple: try to pass on a 'cold' number to your opponent.