Legendre Symbol


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Legendre Symbol

 

the symbol (a/p\ which is equal to 1 if the integer a is a quadratic residue of the odd prime p and is equal to – 1 if a is a quadratic nonresidue of p. The symbol was introduced by A. Legendre in 1785.

References in periodicals archive ?
As is well known, for two odd prime numbers p and q, the Legendre symbol (p/q) describes the decomposition law of q in the quadratic extension Q([square root of p])/Q.
Redei ([R]) introduced a certain triple symbol with the intension of a generalization of the Legendre symbol and Gauss' genus theory.
denotes the Legendre symbol, then x + y could never take the value 0.
Suppose that p is a prime such that (d/p) = 1, where (d/p) is the Legendre symbol. When the norm N([[epsilon].sub.d]) = ([m.sup.2] - d[n.sup.2])/4 = -1, Sun [3] proposed that [[epsilon].sub.d] is quadratic residue mod p if and only if p is represented by one class in the set
where (x=p) denotes the Legendre symbol and m = (p + 1)=2.
where [[chi].sub.2] denotes the Legendre symbol modulo p, [[chi].sub.4] denotes the non-principal character mod 4, and L(1, [[chi].sub.2][[chi].sub.4]) denotes the Dirichlet L-function corresponding to character [[chi].sub.2][[chi].sub.4] mod 4p.
Sun["Quartic residues and binary qudratic forms", Journal of Number Theory, 2005, 113(1)] conjectured that: Let p and q be the primes of the form 4k + 1 such that (p/q) = 1, where (p/q) is the Legendre symbol, then [h.sub.4](-4pq) = [h.sub.4](-64pq) = h(-4pq)=8.
[2,conjecture 8.4] Let p and q be primes of the form 4k+1 such that ( p q ) = 1,where ( p q ) is the Legendre symbol. Then h4([x.sup.-1]4pq) = h4([x.sup.-1]64pq) = h([x.sup.-1]4pq)=8.