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import FormalConjecturesUtilErdős Problem 1141
References:
[APSSV26b] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics, probability and number theory II. arXiv:2604.06609 (2026).
[Or26] Y. Oriike, Lean formalisation of Erdős problem 1141 (2026)
[Po17] P. Pollack, Bounds for the first several prime character nonresidues. Proc. Amer. Math. Soc. (2017), 2815--2826.
[Me1874] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math. (1874), 46--62.
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).
open Nat Setnamespace Erdos1141The cutoff $m^{1/4 + \varepsilon}$ of Theorem 1.3 of [Po17].
noncomputable def residuePrimeUpperBound (m : ℕ) (ε : ℝ) : ℝ := (m : ℝ) ^ ((1 / 4 : ℝ) + ε)The primes $\ell \leq m^{1/4+\varepsilon}$ with $\chi(\ell) = 1$.
This does not require $\chi$ to be quadratic. That hypothesis belongs to the theorem below, as it does in the paper.
noncomputable def residuePrimesUpTo (m : ℕ) (χ : DirichletCharacter ℂ m) (ε : ℝ) : Finset ℕ := m:ℕχ:DirichletCharacter ℂ mε:ℝ⊢ Finset ℕ
classical
All goals completed! 🐙Theorem 1.3 of [Po17]: for a quadratic character to a large enough modulus, the primes below $m^{1/4+\varepsilon}$ on which the character is $1$ outnumber any fixed power of $\log m$.
@[category research solved, AMS 11]
theorem erdos_1141.variants.pollack_1_3 (ε A : ℝ) (hε : 0 < ε) (hA : 0 < A) :
∃ m₀ : ℕ, ∀ m : ℕ, m₀ < m → ∀ χ : DirichletCharacter ℂ m, MulChar.IsQuadratic χ →
Real.log m ^ A ≤ ((residuePrimesUpTo m χ ε).card : ℝ) := ε:ℝA:ℝhε:0 < εhA:0 < A⊢ ∃ m₀,
∀ (m : ℕ),
m₀ < m → ∀ (χ : DirichletCharacter ℂ m), MulChar.IsQuadratic χ → Real.log ↑m ^ A ≤ ↑(residuePrimesUpTo m χ ε).card
All goals completed! 🐙Mertens' third theorem [Me1874], in the weakened form the linked proof assumes: the product over the primes up to $n$ of $1 - 1/p$ is at least $1/(3\log n)$.
The true asymptotic is $e^{-\gamma}/\log n$, and $e^{-\gamma} > 1/3$, so this bound is weaker than the theorem and is what the deduction needs.
@[category research solved, AMS 11]
theorem erdos_1141.variants.mertens_third (n : ℕ) (hn : 3 ≤ n) :
1 / (3 * Real.log n) ≤ ∏ p ∈ (Finset.range (n + 1)).filter Nat.Prime, (1 - 1 / (p : ℝ)) := n:ℕhn:3 ≤ n⊢ 1 / (3 * Real.log ↑n) ≤ ∏ p ∈ Finset.range (n + 1) with Nat.Prime p, (1 - 1 / ↑p)
All goals completed! 🐙The property that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$.
def Erdos1141Prop (n : ℕ) : Prop :=
∀ k, k ^ 2 < n → Coprime n k → (n - k ^ 2).Primeinstance (n : ℕ) : Decidable (Erdos1141Prop n) :=
decidable_of_iff (∀ k ≤ .sqrt (n - 1), Coprime n k → (n - k ^ 2).Prime) <| n:ℕ⊢ (∀ k ≤ (n - 1).sqrt, n.Coprime k → Nat.Prime (n - k ^ 2)) ↔ Erdos1141Prop n
cases n with
⊢ (∀ k ≤ (0 - 1).sqrt, Coprime 0 k → Nat.Prime (0 - k ^ 2)) ↔ Erdos1141Prop 0 All goals completed! 🐙
n':ℕ⊢ (∀ k ≤ (n' + 1 - 1).sqrt, (n' + 1).Coprime k → Nat.Prime (n' + 1 - k ^ 2)) ↔ Erdos1141Prop (n' + 1)
All goals completed! 🐙Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$?
In [Va99] it is asked whether $968$ is the largest integer with this property, but this is an error, since for example $968-9=7\cdot 137$.
The list of $n$ satisfying the given property is [A214583] in the OEIS. The largest known such $n$ is $1722$.
The answer is negative: [APSSV26b] proves a stronger finiteness theorem, deducing it from Pollack [Po17]. Oriike [Or26] formalised the deduction in Lean.
The linked proof is the deduction and not the whole result. It declares Theorem 1.3 of [Po17] and
Mertens' third theorem as axioms, so it is marked conditional and names both.
@[category research solved, AMS 11,
conditional formal_proof using lean4 at
"https://github.com/yuta0x89/ErdosProblems/blob/a1319f732cdee5140faf47d984e2c451c1184803/Erdos1141.lean"
assuming erdos_1141.variants.pollack_1_3 erdos_1141.variants.mertens_third]
theorem erdos_1141 :
answer(False) ↔ Infinite { n | Erdos1141Prop n } := ⊢ False ↔ Infinite ↑{n | Erdos1141Prop n}
All goals completed! 🐙@[category test, AMS 11]
example : ¬ Erdos1141Prop 968 := ⊢ ¬Erdos1141Prop 968
All goals completed! 🐙@[category test, AMS 11]
example : Erdos1141Prop 1722 := ⊢ Erdos1141Prop 1722
All goals completed! 🐙end Erdos1141