Legendre symbol (number theory)

The symbol of Legendre , a theme with its own name in number theory , is linked to the topic of quadratic residuals.

Summary

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• 1 Definition
• 2 Properties
• 3 Examples
• 4 Sources
• 5 References
• 6 See also

Definition

If p represents a prime greater than two and a coprime of p, the Legendre symbol – which is read: “symbol of a with respect to p” – is defined = 1 if a is a quadratic remainder, = -1 if a is a non quadratic residual modulo p. ^[one]

It naturally follows that (a / p) ← → a ^p-1/2 (mod p).

Properties

1. a congruent to b (mod p) implies that (a / p) = (b / p). Which results from the fact that the numbersof the same class are simultaneously or not quadratic remainders.
2. (1 / p) = 1; since 1 = 1 ²and, therefore, 1 is a quadratic remainder.
3. (-1 / p) = (-1) ^p-1/2
4. (a ²/2) = 1
5. (ab … k / p) = (a / p) (b / p) … (k / p)

Examples

1. (-1/11) = -1, (2/11) = -1, (-2/11) = + 1, (3/11) = + 1
2. (-1/13) = +1, (2/13) = -1, (-2/13) = – 1, (3/11) = + 1
3. (-1/17) = +1, (2/17) = +1, (-2/13) = + 1, (3/11) = -1