Jacobi Symbol


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Jacobi Symbol

 

the symbol (n/P), representing a generalization of the Legendre symbol in the case of the composite modulus P. It was introduced by K. Jacobi in 1837. (SeeQUADRATIC RESIDUE.)

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then |[N.sub.0](m, n, d)| = 1/8 h (-4[delta][(n, d).sup.2]d), where [delta](n, d) [member of] {1, 2, 4, 8} is given by table 4 of [3] and [(*/a).sub.4] is the quartic Jacobi symbol. In this paper, we make some numerical evidence to support this conjecture, then pose a stronger version of it.
Keywords Quartic residue, quartic Jacobi symbol, binary quadratic form, class number.
In section 2 we recall and state some basic facts concerning quartic residue characters and qudratic forms, which are necessary for computing the quartic Jacobi symbol. In section 3, using Theorem 8.3 [3] we describe a procedure for searching d [member of] [3, 500] and tabulate the two sets [N.sub.0](m, n, d) and [N.sub.1](m, n, d) for d [member of] [51, 105], then pose a stronger version of conjecture 1.
For [alpha] [member of] Z[i] such that ([alpha], [pi]) = 1, the quartic Jacobi symbol is defined by [([alpha]/[pi]).sub.4] = [([alpha]/[[pi].sub.1]).sub.4]...