English
For odd prime p and odd a with a not divisible by p, the Legendre symbol (p|a) equals (-1) raised to the sum ∑ x ∈ Ico 1 (p/2).succ, x*a/p.
Русский
Для нечетного простого p и нечетного a, не делящегося на p, символ Лежандра (p|a) равен (-1) в степени суммы ∑ x ∈ Ico 1 (p/2).succ, x*a/p.
LaTeX
$$$\operatorname{legendreSym} p a = (-1)^{\sum_{x ∈ Ico 1 (p / 2).succ} x * a / p}$$$
Lean4
/-- The symbol `J(1 | b)` has the value `1`. -/
@[simp]
theorem one_left (b : ℕ) : J(1 | b) = 1 :=
List.prod_eq_one fun z hz => by
let ⟨p, hp, he⟩ := List.mem_pmap.1 hz
rw [← he, legendreSym.at_one]