Legendre symbol evaluator

Properties of the Legendre symbol

p is an odd prime, a ≢ 0 (mod p) and b ≢ 0 (mod p), and q ≠ p is an odd prime.

Congruent numbers

If a ≡ b (mod p) then (a | p) = (b | p)

Square numbers

(a² | p) = 1

Composite numbers

(ab | p) = (a | p) (b | p)

Quadratic character of -1

(-1 | p) = (-1)^{(p - 1) / 2}

Quadratic character of 2

(2 | p) = (-1)^{(p2 - 1) / 8}

Law of quadratic reciprocity (1)

If p ≢ 1 (mod 4) or q ≢ 1 (mod 4) then (p | q) = (q | p)

Law of quadratic reciprocity (2)

If p ≢ q ≢ 3 (mod 4) then (p | q) = -(q | p)

Evaluator

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By Sean McGivern