Smooth gaps between primes

DefinitionDifference-smooth sets of integers#✓fully proved

Let \(y\) be a natural number. A finite set \(H\) of integers is difference \(y\)-smooth if for all distinct \(h, h' \in H\) every prime factor of \(h - h'\) is at most \(y\).

def IsDifferenceSmooth (y : ℕ) (H : Finset ℤ) : Prop :=
  ∀ a ∈ H, ∀ b ∈ H, a ≠ b → (a - b).natAbs ∈ Nat.smoothNumbers (y + 1)

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Definition 3.

Used by 3

Used byAdmissible difference-smooth sets of every size · One-element sets are difference smooth · Admissible sets are not difference smooth at small bounds

TheoremOne-element sets are difference smooth#✓fully proved

For every natural number \(y\) and every integer \(a\), the set \(\{a\}\) is difference \(y\)-smooth.

theorem isDifferenceSmooth_singleton (y : ℕ) (a : ℤ) : IsDifferenceSmooth y {a}

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Definition 3.

Uses 1

Statement usesDifference-smooth sets of integers

TheoremInfinitely many prime pairs with \(47\)-smooth difference#●ready to prove

There are infinitely many pairs of primes \(p > q\) such that every prime factor of \(p - q\) is at most \(47\).

public theorem_wanted infinite_prime_pairs_smooth_47_gap :
    {pq : ℕ × ℕ | pq.1.Prime ∧ pq.2.Prime ∧ pq.2 < pq.1 ∧
      pq.1 - pq.2 ∈ Nat.smoothNumbers 48}.Infinite

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Theorem 1.

TheoremMaynard–Tao theorem#●ready to prove

For every natural number \(m \ge 2\) there is a natural number \(k\) such that for every admissible finite set \(H\) of at least \(k\) integers there are infinitely many integers \(n\) for which \(n + h\) is a (positive) prime for at least \(m\) elements \(h \in H\).

public theorem_wanted maynard_tao_admissible_tuples (m : ℕ) (hm : 2 ≤ m) :
    ∃ k : ℕ, ∀ H : Finset ℤ, IsAdmissibleFinset H → k ≤ H.card →
      {n : ℤ | m ≤ (H.filter (fun h => (n + h).toNat.Prime)).card}.Infinite

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Theorem 2, citing J. Maynard, Small gaps between primes, Ann. of Math. 181 (2015).

Uses 1

Statement usesAdmissible finite sets of integers

TheoremPrime \(m\)-tuples with smooth gaps#●ready to prove

Let \(m \ge 2\) and \(k\) be natural numbers such that every admissible finite set \(H\) of at least \(k\) integers has infinitely many integers \(n\) for which \(n + h\) is a (positive) prime for at least \(m\) elements \(h \in H\). Then there are infinitely many \(m\)-tuples of primes \(p_1 < p_2 < \dots < p_m\) such that for all \(j < i\) every prime factor of \(p_i - p_j\) is at most \(k\).

Equivalently, every gap is \(y\)-smooth for \(y\) the largest prime at most \(k\); the paper applies this with the admissible values \(k_m\) of the Maynard–Tao theorem listed in its Table 1.

public theorem_wanted infinite_prime_tuples_smooth_gaps (m k : ℕ) (hm : 2 ≤ m)
    (hk : ∀ H : Finset ℤ, IsAdmissibleFinset H → k ≤ H.card →
      {n : ℤ | m ≤ (H.filter (fun h => (n + h).toNat.Prime)).card}.Infinite) :
    {P : Fin m → ℕ | (∀ i, (P i).Prime) ∧
      ∀ i j, j < i → P j < P i ∧ P i - P j ∈ Nat.smoothNumbers (k + 1)}.Infinite

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Theorem 3.

Uses 1

Statement usesAdmissible finite sets of integers

TheoremPrime pairs with \(5\)-smooth difference under Elliott–Halberstam#●ready to prove

If the Elliott–Halberstam conjecture holds, then there are infinitely many pairs of primes \(p > q\) such that every prime factor of \(p - q\) is at most \(5\).

public theorem_wanted infinite_prime_pairs_smooth_5_gap_of_elliottHalberstam :
    WantedExt.NumberTheory.Polymath8b.ElliottHalberstam →
    {pq : ℕ × ℕ | pq.1.Prime ∧ pq.2.Prime ∧ pq.2 < pq.1 ∧
      pq.1 - pq.2 ∈ Nat.smoothNumbers 6}.Infinite

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Theorem 4.

Uses 1

Statement usesElliott–Halberstam conjecture (dyadic form)

TheoremAdmissible difference-smooth sets of every size#●ready to prove

For every natural number \(k\) there is an admissible set \(H\) of exactly \(k\) integers that is difference \(k\)-smooth: for all distinct \(h, h' \in H\), every prime factor of \(h - h'\) is at most \(k\).

public theorem_wanted exists_admissible_differenceSmooth (k : ℕ) :
    ∃ H : Finset ℤ, H.card = k ∧ IsAdmissibleFinset H ∧ IsDifferenceSmooth k H

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Theorem 5.

Uses 2

Statement usesAdmissible finite sets of integers · Difference-smooth sets of integers

TheoremAdmissible sets are not difference smooth at small bounds#●ready to prove

Let \(H\) be an admissible finite set of \(k \ge 2\) integers, let \(z \le k\) be a prime and let \(\ell < z\) be a natural number. Then \(H\) is not difference \(\ell\)-smooth. Equivalently, \(H\) is not difference \(\ell\)-smooth for any \(\ell\) less than the largest prime at most \(k\).

public theorem_wanted not_isDifferenceSmooth_of_isAdmissibleFinset (H : Finset ℤ)
    (hk : 2 ≤ H.card) (hH : IsAdmissibleFinset H) (z : ℕ) (hz : z.Prime) (hzk : z ≤ H.card)
    (ℓ : ℕ) (hℓ : ℓ < z) : ¬ IsDifferenceSmooth ℓ H

Sources

• Carol Wu, On Smooth Gaps between Primes Using the Maynard-Tao Sieve, INTEGERS 25 (2025), Article A44, Proposition 1.

Uses 2

Statement usesAdmissible finite sets of integers · Difference-smooth sets of integers

2 fully proved6 ready to prove

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