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The difference between any perfect square and its predecessor is given by the identity n 2 − (n − 1) 2 = 2n − 1.Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is, n 2 = (n − 1) 2 + (n − 1) + n.
Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion.In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field (/).
Incomplete ultimate tic-tac-toe game (the large 'X's and 'O's represent small boards that are won by that player). The previous move was O playing in the center square of the bottom-right board, forcing X to play their next move in smaller board located in the center of the larger board (highlighted in blue).
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.It is one of many closures in mathematics.