generalized Jacobi symbol
generalized Legendre symbol
positive odd integer
The KroneckerSymbol(a, n) command computes the Kronecker symbol of a and n.
The alternative calling sequences, JacobiSymbol(a, m) and LegendreSymbol(a, m), have return values equal to KroneckerSymbol(a, m), but m must be a positive odd integer.
The Legendre symbol is typically defined only for second arguments that are prime, but due to primality checking being expensive, here LegendreSymbol is an alias of JacobiSymbol.
If n is equal to u⁢∏i=1k⁡pidi where u is −1 or 1 and the pi are distinct primes, then the Kronecker symbol an is given by au∏i=1kapidi, where ak is the usual Legendre symbol, except for the following cases.
a0 is equal to 1 if a is equal to 1 or −1. Otherwise it is equal to 0.
a-1 is equal to −1 if a is less than 0. Otherwise it is equal to 1.
a1 is always equal to 1.
a2 is equal to 2a if a is odd. Otherwise it is equal to 0.
22 is congruent to 0 modulo 11.
9 is a quadratic residue modulo 11.
10 is a quadratic non-residue modulo 11.
11 is congruent to 2 modulo 3 and is a quadratic non-residue modulo 3.
The NumberTheory[KroneckerSymbol], NumberTheory[JacobiSymbol] and NumberTheory[LegendreSymbol] commands were introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
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