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Jacobi symbol

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The Jacobi symbol is a generalization of the Legendre symbol. Given two coprime integers and , with the former the product of primes (not necessarily distinct),[1] the Jacobi symbol is

where is the Legendre symbol. Here are two examples with :

There is no problem with confusing the Legendre and Jacobi symbols; one has a prime for the second argument, the other a composite. In fact, at least one computer algebra system (Wolfram Mathematica) does not offer a separate LegendreSymbol[a, p] command,[2] instead "overloading" JacobiSymbol[n, m].

However, the notation remains confusing because it looks like a fraction: the examples above could be misunderstood to mean that 2 fifteenths is the same thing as 4 fifteenths and 1, or that 2 forty-fifths is the same thing as 8 forty-fifths and –1.

  1. By we're referring to Omega(n), number of prime factors of n (with multiplicity).
  2. But one can certainly define it if one feels that strongly about it.