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import FormalConjecturesUtilNumber of squares $\bmod n$
The number of squares modulo $n$. This is the cardinality of the set ${k^2 \bmod n \mid k \in {0, 1, \dots, n-1}}$.
References:
namespace OeisA224The number of squares modulo $n$.
def a (n : ℕ) : ℕ :=
if n = 0 then 1
else
Finset.card ((Finset.range n).image (fun k : ℕ => k ^ 2 % n))
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 2 := ⊢ a 3 = 2 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 2 := ⊢ a 4 = 2 All goals completed! 🐙$n^2 \equiv 1 \pmod{a(n)(a(n)-1)}$ if and only if $n$ is an odd prime.
Thomas Ordowski, Jun 08 2017
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 1 < n) :
(n.Prime ∧ n ≠ 2) ↔ n ^ 2 ≡ 1 [MOD a n * (a n - 1)] := n:ℕhn:1 < n⊢ Nat.Prime n ∧ n ≠ 2 ↔ n ^ 2 ≡ 1 [MOD a n * (a n - 1)]
All goals completed! 🐙end OeisA224