The Jacobi Symbol
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چکیده
is only defined when the bottom is an odd prime. You can extend the definition to allow an odd positive number on the bottom using the Jacobi symbol. Most of the properties of Legendre symbols go through for Jacobi symbols, which makes Jacobi symbols very convenient for computation. We’ll see, however, that there is a price to pay for the greater generality: Euler’s formula no longer works, and we lose part of the connection between the value of a symbol and the solvability of the corresponding quadratic congruence. Definition. Let p, q ∈ Z, where (p, q) = 1 and q is a product of odd primes: q = q1q2 · · · qn. (The qi need not be distinct.) The Jacobi symbol
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The Jacobi symbol satisfies many formulas that the Legendre symbol does, such as these: for a, b ∈ Z and odd m,n ∈ Z+, (1) a ≡ b mod n⇒ ( a n) = ( b n), (2) ( n ) = ( a n)( b n), (3) (−1 n ) = (−1) (n−1)/2 and ( 2 n) = (−1) (n2−1)/8, (4) ( n m) = (−1) (n ). But there is one basic rule about Legendre symbols that is not listed above for the Jacobi symbol: an analogue of Euler’s congruence a(p−1)...
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