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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 (/).
This can be seen in the following tables, the left of which shows Newton's method applied to the above f(x) = x + x 4/3 and the right of which shows Newton's method applied to f(x) = x + x 2. The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1 ...
7 1 Kuala Selangor District: 1,941 12 162 9 3 Hulu Langat District: 21,006 17 1,236 16 1 Hulu Selangor District: 2,027 9 225 9 0 Sabak Bernam District: 2,009 10 201 3 7 Gombak District: 9,181 8 1,148 8 0 Petaling District: 38,152 23 1,659 23 0 Sepang District: 4,404 7 629 6 1 Total: 107,444: 115 934: 99 16