Fractional parts of multiples of primes

DefinitionWell-distributed sequences#✓fully proved

A sequence \((x_m)_{m \in \mathbb{N}}\) of real numbers is well-distributed if for every \(\varepsilon > 0\) there is a \(K\) such that for all \(k \ge K\), all \(n \in \mathbb{N}\) and all real numbers \(a, b\) with \(0 \le a \le b \le 1\), the number \(N\) of indices \(m\) with \(n < m \le n + k\) and \(x_m \in [a, b)\) satisfies \(|N - (b - a)k| < \varepsilon k\).

No condition \(x_m \in [0, 1)\) is imposed, so finite changes to a sequence do not affect this property.

def WellDistributed (x : ℕ → ℝ) : Prop :=
  ∀ ε : ℝ, 0 < ε → ∀ᶠ k : ℕ in Filter.atTop, ∀ n : ℕ,
    ∀ a b : ℝ, 0 ≤ a → a ≤ b → b ≤ 1 →
      let I := Set.Ico a b
      let count := (Finset.Ioc n (n + k)).filter (fun m ↦ x m ∈ I)
      abs ((count.card : ℝ) - (b - a) * k) < ε * k

Sources

• EP-997 (Erdős Problem #997) [Erdős Problems].

Used by 1

Used byStatement of Erdős Problem 997

DefinitionFractional parts of multiples of the primes#✓fully proved

For a real number \(\alpha\), this is the sequence taking \(n \in \mathbb{N}\) to the fractional part \(\{\alpha p_n\} = \alpha p_n - \lfloor \alpha p_n \rfloor\), where \(p_0 = 2, p_1 = 3, p_2 = 5, \dots\) are the primes in increasing order.

noncomputable def primeFract (α : ℝ) (n : ℕ) : ℝ :=
  Int.fract (α * (Nat.nth Nat.Prime n : ℝ))

Sources

• EP-997 (Erdős Problem #997) [Erdős Problems].

Used by 1

Used byStatement of Erdős Problem 997

DefinitionStatement of Erdős Problem 997#✓fully proved

For every real number \(\alpha\), the sequence \(n \mapsto \{\alpha p_n\}\) of fractional parts of \(\alpha\) times the primes \(p_0 = 2, p_1 = 3, p_2 = 5, \dots\) is not well-distributed.

noncomputable def Statement : Prop :=
  ∀ α : ℝ, ¬ WellDistributed (primeFract α)

Sources

• EP-997 (Erdős Problem #997) [Erdős Problems].

Uses 2 · Used by 1

Statement usesWell-distributed sequences · Fractional parts of multiples of the primes

Used byErdős Problem 997 holds

TheoremErdős Problem 997 holds#●ready to prove

For every real number \(\alpha\), the sequence \(n \mapsto \{\alpha p_n\}\) of fractional parts of \(\alpha\) times the primes \(p_0 = 2, p_1 = 3, p_2 = 5, \dots\) is not well-distributed.

public theorem_wanted Statement_holds : Statement

Sources

• Boris Alexeev, Mark Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant, Short proofs in combinatorics and number theory, arXiv:2603.29961 (2026), https://arxiv.org/abs/2603.29961. Resolved true: Alexeev, Putterman, Sawhney, Sellke, and Valiant proved that the prime-indexed fractional-part sequence is never well-distributed.

Uses 1

Statement usesStatement of Erdős Problem 997

3 fully proved1 ready to prove

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