Weakly prime-additive numbers

DefinitionSums of powers of distinct prime divisors#✓fully proved

For natural numbers \(n\) and \(t\), \(n\) is a sum of positive powers of \(t\) distinct prime divisors of \(n\) when there are a set \(s\) of \(t\) distinct primes, each dividing \(n\), and positive integers \(e_p\) for \(p \in s\) such that \(n = \sum_{p \in s} p^{e_p}\).

public def IsPrimeDivisorPowerSum (n t : ℕ) : Prop :=
  ∃ s : Finset ℕ, s.card = t ∧ (∀ p ∈ s, p.Prime ∧ p ∣ n) ∧
    ∃ e : ℕ → ℕ, (∀ p ∈ s, 0 < e p) ∧ n = ∑ p ∈ s, p ^ e p

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Section 1, Definition (after Erdős and Hegyvári).

Used by 5

Used byThirty is a sum of powers of three of its prime divisors · Weakly prime-additive numbers · Thirty is weakly prime-additive · Weakly prime-additive length · The length is at most the size of any representation

DefinitionWeakly prime-additive numbers#✓fully proved

A natural number \(n\) is weakly prime-additive if it has at least two distinct prime divisors and is a sum \(n = \sum_{p \in s} p^{e_p}\) of positive powers \(p^{e_p}\), \(e_p \ge 1\), over some set \(s\) of distinct prime divisors of \(n\).

public def IsWeaklyPrimeAdditive (n : ℕ) : Prop :=
  2 ≤ n.primeFactors.card ∧ ∃ t, IsPrimeDivisorPowerSum n t

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Section 1, Definition.

Uses 1 · Used by 4

Statement usesSums of powers of distinct prime divisors

Used byFang–Chen theorem on shortest weakly prime-additive multiples · Weakly prime-additive multiples of length four · Weakly prime-additive multiples of length at most four · Thirty is weakly prime-additive

DefinitionWeakly prime-additive length#✓fully proved

For a natural number \(n\), the length \(\kappa_n\) is the least \(t\) such that \(n\) is a sum of positive powers of \(t\) distinct prime divisors of \(n\), and \(\kappa_n = 0\) when \(n\) has no such representation.

public noncomputable def weaklyPrimeAdditiveLength (n : ℕ) : ℕ :=
  sInf {t | IsPrimeDivisorPowerSum n t}

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Section 1, Definition.

Uses 1 · Used by 4

Statement usesSums of powers of distinct prime divisors

Used byFang–Chen theorem on shortest weakly prime-additive multiples · Weakly prime-additive multiples of length four · Weakly prime-additive multiples of length at most four · The length is at most the size of any representation

TheoremThirty is a sum of powers of three of its prime divisors#✓fully proved

The number \(30\) is a sum of positive powers of three distinct prime divisors of \(30\), namely \(30 = 2 + 3 + 5^2\).

public theorem isPrimeDivisorPowerSum_thirty : IsPrimeDivisorPowerSum 30 3

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Section 1, Definition.

Uses 1 · Used by 1

Statement usesSums of powers of distinct prime divisors

Used byThirty is weakly prime-additive

TheoremThirty is weakly prime-additive#✓fully proved

The number \(30\) is weakly prime-additive.

Indeed, \(30\) has the distinct prime divisors \(2\), \(3\), \(5\) and \(30 = 2 + 3 + 5^2\).

public theorem isWeaklyPrimeAdditive_thirty : IsWeaklyPrimeAdditive 30

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Section 1, Definition.

Uses 3

Statement usesWeakly prime-additive numbers

Proof usesSums of powers of distinct prime divisors · Thirty is a sum of powers of three of its prime divisors

TheoremThe length is at most the size of any representation#✓fully proved

For natural numbers \(n\) and \(t\), if \(n\) is a sum of positive powers of \(t\) distinct prime divisors of \(n\), then the length \(\kappa_n\) of \(n\) is at most \(t\).

public theorem weaklyPrimeAdditiveLength_le {n t : ℕ} (h : IsPrimeDivisorPowerSum n t) :
    weaklyPrimeAdditiveLength n ≤ t

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Section 1, Definition (minimality of κ_n).

Uses 2

Statement usesSums of powers of distinct prime divisors · Weakly prime-additive length

TheoremWeakly prime-additive multiples of length four#●ready to prove

Assume \((\ast)\): for all natural numbers \(a\), \(f\), \(g\) with \(1 \le a \le f\), \(\gcd(a, f) = 1\) and \(4 \mid f\), and \(g\) an odd prime dividing \(f\) with Jacobi symbol \(\left(\frac{g}{a}\right) = -1\), there is a prime \(\ell \equiv a \pmod f\) such that \(g\) is a primitive root modulo \(\ell\). Then for every positive integer \(m\) there are infinitely many weakly prime-additive numbers \(n\) with \(m \mid n\) and length \(\kappa_n = 4\).

The paper states \((\ast)\) with the Kronecker symbol; since \(a\) is coprime to \(f\) and \(4 \mid f\), \(a\) is odd, so it is the Jacobi symbol.

public theorem_wanted infinite_weaklyPrimeAdditive_length_four_dvd
    (hstar : ∀ a f g : ℕ, 1 ≤ a → a ≤ f → Nat.Coprime a f → 4 ∣ f → g.Prime → Odd g →
      g ∣ f → jacobiSym g a = -1 →
      ∃ p : ℕ, p.Prime ∧ p ≡ a [MOD f] ∧ IsPrimitiveRoot (g : ZMod p) (p - 1))
    (m : ℕ) (hm : 0 < m) :
    {n : ℕ | m ∣ n ∧ IsWeaklyPrimeAdditive n ∧ weaklyPrimeAdditiveLength n = 4}.Infinite

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Theorem 1.

Uses 2

Statement usesWeakly prime-additive numbers · Weakly prime-additive length

TheoremPrimes completing a four-term prime-power divisibility#●ready to prove

Assume \((\ast)\): for all natural numbers \(a\), \(f\), \(g\) with \(1 \le a \le f\), \(\gcd(a, f) = 1\) and \(4 \mid f\), and \(g\) an odd prime dividing \(f\) with Jacobi symbol \(\left(\frac{g}{a}\right) = -1\), there is a prime \(\ell \equiv a \pmod f\) such that \(g\) is a primitive root modulo \(\ell\). Let \(p\), \(q\), \(r\) be distinct odd primes, at least one of which is congruent to \(3\) or \(5\) modulo \(8\). Then there are infinitely many primes \(s\), distinct from \(p\), \(q\) and \(r\), for each of which there are infinitely many quadruples \((e_1, e_2, e_3, e_4)\) of positive integers with \(pqrs \mid p^{e_1} + q^{e_2} + r^{e_3} + s^{e_4}\).

public theorem_wanted infinite_primes_dvd_four_prime_power_sum
    (hstar : ∀ a f g : ℕ, 1 ≤ a → a ≤ f → Nat.Coprime a f → 4 ∣ f → g.Prime → Odd g →
      g ∣ f → jacobiSym g a = -1 →
      ∃ p : ℕ, p.Prime ∧ p ≡ a [MOD f] ∧ IsPrimitiveRoot (g : ZMod p) (p - 1))
    (p q r : ℕ) (hp : p.Prime) (hq : q.Prime) (hr : r.Prime)
    (hp2 : Odd p) (hq2 : Odd q) (hr2 : Odd r) (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r)
    (hmod : p % 8 = 3 ∨ p % 8 = 5 ∨ q % 8 = 3 ∨ q % 8 = 5 ∨ r % 8 = 3 ∨ r % 8 = 5) :
    {s : ℕ | s.Prime ∧ s ≠ p ∧ s ≠ q ∧ s ≠ r ∧
      {x : ℕ × ℕ × ℕ × ℕ | 0 < x.1 ∧ 0 < x.2.1 ∧ 0 < x.2.2.1 ∧ 0 < x.2.2.2 ∧
        p * q * r * s ∣ p ^ x.1 + q ^ x.2.1 + r ^ x.2.2.1 + s ^ x.2.2.2}.Infinite}.Infinite

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Theorem 2.

TheoremFang–Chen theorem on shortest weakly prime-additive multiples#●ready to prove

For every positive integer \(m\), there are infinitely many weakly prime-additive numbers \(n\) with \(m \mid n\) and length \(\kappa_n = 3\) if and only if \(8 \nmid m\).

public theorem_wanted infinite_shortest_weaklyPrimeAdditive_dvd_iff (m : ℕ) (hm : 0 < m) :
    {n : ℕ | m ∣ n ∧ IsWeaklyPrimeAdditive n ∧ weaklyPrimeAdditiveLength n = 3}.Infinite ↔
      ¬ 8 ∣ m

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Theorem 5 (Fang and Chen, Corollary 1.1).

Uses 2

Statement usesWeakly prime-additive numbers · Weakly prime-additive length

TheoremWeakly prime-additive multiples of length at most four#●ready to prove

Assume \((\ast)\): for all natural numbers \(a\), \(f\), \(g\) with \(1 \le a \le f\), \(\gcd(a, f) = 1\) and \(4 \mid f\), and \(g\) an odd prime dividing \(f\) with Jacobi symbol \(\left(\frac{g}{a}\right) = -1\), there is a prime \(\ell \equiv a \pmod f\) such that \(g\) is a primitive root modulo \(\ell\). Then for every positive integer \(m\) there are infinitely many weakly prime-additive numbers \(n\) with \(m \mid n\) and length \(\kappa_n \le 4\).

public theorem_wanted infinite_weaklyPrimeAdditive_length_le_four_dvd
    (hstar : ∀ a f g : ℕ, 1 ≤ a → a ≤ f → Nat.Coprime a f → 4 ∣ f → g.Prime → Odd g →
      g ∣ f → jacobiSym g a = -1 →
      ∃ p : ℕ, p.Prime ∧ p ≡ a [MOD f] ∧ IsPrimitiveRoot (g : ZMod p) (p - 1))
    (m : ℕ) (hm : 0 < m) :
    {n : ℕ | m ∣ n ∧ IsWeaklyPrimeAdditive n ∧ weaklyPrimeAdditiveLength n ≤ 4}.Infinite

Sources

• Wing Hong Leung, On the Weakly Prime-Additive Numbers with Length 4, INTEGERS 24 (2024), Article A9, Theorem 6.

Uses 2

Statement usesWeakly prime-additive numbers · Weakly prime-additive length

6 fully proved4 ready to prove

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