# Legendre Symbol Calculator

Legendre Symbol is a mathematical theoretical function (a/p) with values equivalent to 1, -1 and 0 based on a quadratic character modulo 'p'. Here, let 'p' be an odd prime and 'a' be an arbitrary integer. On a non zero quadratic residue mod 'p' , the value is 1. On a non quadratic residue it is -1 and on zero, it is 0. The symbol is used in law of quadratic reciprocity to simplify notation. Find the legendre symbol with numerator and denominator value.

Legendre Symbol is a mathematical theoretical function (a/p) with values equivalent to 1, -1 and 0 based on a quadratic character modulo 'p'. Here, let 'p' be an odd prime and 'a' be an arbitrary integer. On a non zero quadratic residue mod 'p' , the value is 1. On a non quadratic residue it is -1 and on zero, it is 0. The symbol is used in law of quadratic reciprocity to simplify notation. Find the legendre symbol with numerator and denominator value.

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#### Rules To Find Legendre Symbol

- (a/n) = (b/n) if a = b mod n.
- (1/n) = 1 and (0/n) = 0.
- (2m/n) = (m/n) if n = ±1 mod 8. Otherwise (2m/n) = -(m/n).
- (Quadratic reciprocity) If m and n are both odd, then (m/n) = (n/m) unless both m and n are congruent to 3 mod 4, in which case (m/n) = -(n/m).

### Example:

If the numerator is 30 and denominator is 23,

#### 30 / 23

= 7 / 23 (i.e., 30 mod 23)

= -2 / 7 (i.e., both odd reciprocol multiply with -1 and take mod)

= -1 / 7

= -1