# Legendre Symbol and Quadratic Reciprocity

• January 24th 2010, 09:29 AM
Pythagoras_barrel
The question asks to define a Legendre symbol and use the Law of Quadratic Reciprocity to determine whether 135 is a quadratic residue modulo 283.

I already know how to define a Legendre symbol but I'm unsure of how to use the Law of Quadratic Reciprocity to answer this question? :confused:
• January 25th 2010, 08:31 AM
tonio
Quote:

Originally Posted by Pythagoras_barrel
The question asks to define a Legendre symbol and use the Law of Quadratic Reciprocity to determine whether 135 is a quadratic residue modulo 283.

I already know how to define a Legendre symbol but I'm unsure of how to use the Law of Quadratic Reciprocity to answer this question? :confused:

I really don't know what you mean by "defining a Legendre symbol", but since $3=283=3\!\!\!\pmod 4$ , we obtain by the multiplicativity of the Legendre symbol and by the quadratic reciprocity theorem $\binom{135}{283}=\binom{27}{283}\binom{5}{283}$ . Now:

** $\binom{5}{283}=\binom{283}{5}=\binom{3}{5}=-1$

Tonio

** $\binom{3}{283}=-\binom{283}{3}=-\binom{1}{3}=-1$ , and then $27=3^2\cdot 3$ is the product of a quadratic residue (9) and a non-quadratic residue (3), and thus it itself is a

non-quad. res., so $\binom{27}{283}=-1$

In short, 135 is the product of two non-quad. res. and thus it is a quad. residue. After some calculation, $237^2=135\!\!\!\pmod{283}$