/-
Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module
public import FormalConjecturesUtil
@[expose] public sectionnamespace Erdos884The sum $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i}$ over all pairs of divisors $d_i < d_j$ of $n$.
noncomputable abbrev sumDivisorInvPairwiseDifference (n : ℕ) : ℝ :=
∑ j : Fin n.divisors.card, ∑ i : Fin j,
(1 : ℚ) / (Nat.nth (· ∣ n) j - Nat.nth (· ∣ n) i )The sum $\sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ over consecutive divisors of $n$.
noncomputable abbrev sumDivisorInvConsecutiveDifference (n : ℕ) : ℝ :=
∑ i : Fin (n.divisors.card - 1),
(1 : ℚ) / (Nat.nth (· ∣ n) (i + 1) - Nat.nth (· ∣ n) i)For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of $n$ in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty$, i.e. $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i} \right)$?
This conjecture has been disproved:
In September 2025, Terence Tao gave a conditional negative answer assuming the prime tuples
conjecture, see erdos_884_false_of_hardy_littlewood for this implication.
Daniel Larsen subsequently gave an unconditional disproof.
def Erdos884Prop : Prop :=
sumDivisorInvPairwiseDifference ≪ 1 + sumDivisorInvConsecutiveDifferenceFor a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of $n$ in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty$, i.e. $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i} \right)$?
This conjecture has been disproved:
In September 2025, Terence Tao gave a conditional negative answer assuming the prime tuples
conjecture, see erdos_884_false_of_hardy_littlewood for this implication.
Daniel Larsen subsequently gave an unconditional disproof.
Reference: erdosproblems.com/884
@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/Jayyhk/erdos-lean/blob/f8a51976fd2e66a52b4928c109fb9ae877a1a507/problems/884/Erdos884.lean"]
theorem erdos_884 :
answer(False) ↔ Erdos884Prop := ⊢ False ↔ Erdos884Prop
All goals completed! 🐙In September 2025, Terence Tao gave a conditional negative answer to Erdős problem 884, disproving it under the assumption of the qualitative Hardy–Littlewood prime tuples conjecture, see Conjecture 1.1 and Theorem 1.1 of Tao25.
The qualitative conjecture states that for every admissible tuple $(h_1, \dotsc, h_k)$ of integers, i.e. one which avoids at least one residue class modulo $p$ for every prime $p$, there are infinitely many natural numbers $n$ such that $n + h_1, \dotsc, n + h_k$ are all prime. Unlike the full Hardy–Littlewood conjecture, it does not give an asymptotic for the number of such $n$.
@[category research solved, AMS 11]
theorem erdos_884_false_of_hardy_littlewood
(hHL : ∀ H : Finset ℤ, (∀ p : ℕ, p.Prime → ∃ a : ℤ, ∀ h ∈ H, ¬ (p : ℤ) ∣ a + h) →
{n : ℕ | ∀ h ∈ H, Prime ((n : ℤ) + h)}.Infinite) :
¬Erdos884Prop := hHL:∀ (H : Finset ℤ), (∀ (p : ℕ), Nat.Prime p → ∃ a, ∀ h ∈ H, ¬↑p ∣ a + h) → {n | ∀ h ∈ H, Prime (↑n + h)}.Infinite⊢ ¬Erdos884Prop
All goals completed! 🐙end Erdos884