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import FormalConjecturesUtilProduct of two consecutive primes modulo the next prime
The sequence is defined by $$a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2),$$ where $\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1)=2$).
References:
namespace OeisA182126$\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1) = 2$).
noncomputable def prime (k : ℕ) : ℕ := Nat.nth Nat.Prime (k - 1)$a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2)$.
noncomputable def a (n : ℕ) : ℕ :=
if n = 0 then 0
else (prime n * prime (n + 1)) % prime (n + 2)All goals completed! 🐙All goals completed! 🐙All goals completed! 🐙All goals completed! 🐙Count of occurrences of value $v$ among $a(1), \dots, a(x)$.
noncomputable def countA (x v : ℕ) : ℕ :=
((Finset.range (x + 1)).filter fun n => 1 ≤ n ∧ a n = v).card$v_0$ is a most frequent value among $a(1), \dots, a(x)$.
def IsMostFrequent (x v₀ : ℕ) : Prop :=
∀ v : ℕ, countA x v ≤ countA x v₀Conjecture: For $x > 10^9$, the most frequent value in $a(n)$, $n=1\dots x$, has form $120k$.
@[category research open, AMS 11]
theorem conjecture1 (x : ℕ) (hx : 10 ^ 9 < x) (v₀ : ℕ) (hv : IsMostFrequent x v₀) :
120 ∣ v₀ := x:ℕhx:10 ^ 9 < xv₀:ℕhv:IsMostFrequent x v₀⊢ 120 ∣ v₀
All goals completed! 🐙Let $b = \mathrm{prime}(n+2) - \mathrm{prime}(n)$ and $c = \mathrm{prime}(n+2) - \mathrm{prime}(n+1)$. Conjecture: for $n > 61$, $a(n) = b \cdot c$.
Charles R Greathouse IV, May 11 2012
@[category research open, AMS 11]
theorem conjecture2 (n : ℕ) (hn : 61 < n) :
let b := prime (n + 2) - prime n
let c := prime (n + 2) - prime (n + 1)
a n = b * c := n:ℕhn:61 < n⊢ let b := prime (n + 2) - prime n;
let c := prime (n + 2) - prime (n + 1);
a n = b * c
All goals completed! 🐙Are 2, 7, 11, 13, 29 the only primes in this sequence?
Hugo Pfoertner, Sep 22 2025
@[category research open, AMS 11]
theorem conjecture3 (n : ℕ) (hn : 0 < n) :
(a n).Prime ↔ a n ∈ ([2, 7, 11, 13, 29] : List ℕ) := n:ℕhn:0 < n⊢ Nat.Prime (a n) ↔ a n ∈ [2, 7, 11, 13, 29]
All goals completed! 🐙end OeisA182126