Calculating The Jacobi Symbol
The above formulas lead to an efficient algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the GCD of two numbers. (This should not be surprising in light of rule 3)).
The "numerator" is reduced modulo the "denominator" using rule 2). Any multiples of 2 are pulled out using rule 4) and calculated using rule 8). The symbol is flipped using rule 6), and the algorithm recurses until the "numerator" is 1 (covered by rule 4)) or 2 (covered by rule 8)), or the "numerator" equals the "denominator" (rule 3)).
Read more about this topic: Jacobi Symbol
Famous quotes containing the words calculating the, calculating, jacobi and/or symbol:
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)
“Sin in this country has been always said to be rather calculating than impulsive.”
—Frank Moore Colby (1865–1925)
“... spinsterhood [is considered to be] an abnormality of small proportions and small consequence, something like an extra finger or two on the body, presumably of temporary duration, and never of any social significance.”
—Mary Putnam Jacobi (1842–1906)
“The truth has never been of any real value to any human being—it is a symbol for mathematicians and philosophers to pursue. In human relations kindness and lies are worth a thousand truths.”
—Graham Greene (1904–1991)