Nonunique Factorization of Abundant Numbers Paul Baginski Fairfield - - PowerPoint PPT Presentation

Nonunique Factorization of Abundant Numbers

Paul Baginski

Fairfield University

Additive Combinatorics / Combinatoire Additive CIRM, Luminy September 8, 2020

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

For each a ≥ 2, the sum of divisors is σ(a) =

• d|a

d Each a ≥ 2 is abundant if σ(a) > 2a perfect if σ(a) = 2a deficient if σ(a) < 2a A = {a ∈ N | a is abundant } = {a ∈ N | σ(a) > 2a} = {12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, . . . , 945, . . .} D = {a ∈ N | a is non-deficient } = {a ∈ N | σ(a) ≥ 2a} = {6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, . . . , 945, . . .}

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A = {a ∈ N | a is abundant } D = {a ∈ N | a is non-deficient } If P ⊆ P is a set of primes, we can localize: AP = {a ∈ A | Supp(a) ⊆ P} DP = {a ∈ D | Supp(a) ⊆ P} Example: A{3,5,7} = {a = 3e15e27e3 | a is abundant} Facts:

1 If |P| = 1, then AP = DP = ∅. 2 If |P| = 2 and 2 /

∈ P, then AP = DP = ∅.

3 As min P → ∞, we must have |P| → ∞ to guarantee AP = ∅

(and DP = ∅).

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A = {a ∈ N | a is abundant } D = {a ∈ N | a is non-deficient } Both A and D are partially ordered by |. The minimal elements of A are called primitive abundant numbers, while the minimal elements of D are called primitive nondeficient numbers. Say primitive for short. Theorem (Dickson 1913) For each r ≥ 2, there are only finitely many odd primitive a ∈ N with |Supp(a)| ≤ r.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

σ is a multiplicative function. Moreover: ∀a, b ≥ 1 bσ(a) < σ(ba) So A and D are both multiplicative subsemigroups of N≥1.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

σ is a multiplicative function. Moreover: ∀a, b ≥ 1 bσ(a) < σ(ba) So A and D are both multiplicative subsemigroups of N≥1. In fact, they are both semigroup ideals (s-ideals) of N≥1. What is their factorization structure?

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Let H = A or H = D. An element a ∈ H is irreducible in H if a cannot be written as a = bc for some b, c ∈ H. Proposition If a ∈ H is primitive, then a is irreducible. Converse is false: 24 is irreducible but not primitive because 12|24. Proposition For any P = {p1, . . . , ps} ⊆ P finite, if HP = ∅, then HP is a divisor-closed subsemigroup of H and H is isomorphic to an additive subsemigroup of Ns. Dickson’s Theorem says HP has only finitely many primitives. So does HP have only finitely many irreducibles?

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Let H = A or H = D. An element a ∈ H is irreducible in H if a cannot be written as a = bc for some b, c ∈ H. Proposition If a ∈ H is primitive, then a is irreducible. Converse is false: 24 is irreducible but not primitive because 12|24. Proposition For any P = {p1, . . . , ps} ⊆ P finite, if HP = ∅, then HP is a divisor-closed subsemigroup of H and H is isomorphic to an additive subsemigroup of Ns. Theorem For any P ⊆ P finite, if HP = ∅, then HP contains infinitely many irreducibles on H.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Factorization into irreducibles is not unique in H: 24 · 24 = 12 · 48

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Factorization into irreducibles is not unique in H: 24 · 24 = 12 · 48 Factorization is not half-factorial: 18 · 20 · 24 = 72 · 120 9453 = 1575 · 535815 The length set of x is L(x) = {ℓ ∈ N | x factors as a product of ℓ irreducibles}

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A semigroup S is bifurcus if every reducible x ∈ S factors as a product of two irreducibles, i.e. min L(x) ≤ 2 for all x ∈ S. Theorem If HP ⊆ qrN for some distinct primes q, r, then HP is bifurcus. Example Let P = {3, 5, 7, 11}. Every nondeficient number with support in P is divisible by 15. Since HP ⊆ 15N, so HP is bifurcus.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A semigroup S is bifurcus if every reducible x ∈ S factors as a product of two irreducibles, i.e. min L(x) ≤ 2 for all x ∈ S. Theorem If HP ⊆ qrN for some distinct primes q, r, then HP is bifurcus. Example Let P = {3, 5, 7, 11}. Every nondeficient number with support in P is divisible by 15. Since HP ⊆ 15N, so HP is bifurcus. However, for P = {3, 5, 7, 11, 13}, HP ⊆ qrN for any q, r ∈ {3, 5, 7, 11, 13}, because a = 33 · 5 · 7, b = 33 · 52 · 11, and c = 32 · 72 · 112 · 133 are elements of HP. So the theorem does not apply to this HP.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A semigroup S is m-furcus if every reducible x ∈ S factors as a product of at most m irreducibles, i.e. min L(x) ≤ m for all x ∈ S. Note: Bifurcus is 2-furcus. If m ≤ n then m-furcus ⇒ n-furcus. Theorem If P is finite and HP = ∅, then HP is m-furcus for some m ≤ |P| + 1.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A semigroup S is m-furcus if every reducible x ∈ S factors as a product of at most m irreducibles, i.e. min L(x) ≤ m for all x ∈ S. Note: Bifurcus is 2-furcus. If m ≤ n then m-furcus ⇒ n-furcus. Theorem If P is finite and HP = ∅, then HP is m-furcus for some m ≤ |P| + 1. Theorem For every m ≥ 2, there exists a ∈ H with min L(a) ≥ m. Corollary For every m ≥ 2, there exists a finite P and an ℓ ≥ m such that HP = ∅ and HP is ℓ-furcus but not (ℓ − 1)-furcus.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A semigroup S is m-furcus if every reducible x ∈ S factors as a product of at most m irreducibles, i.e. min L(x) ≤ m for all x ∈ S. Note: Bifurcus is 2-furcus. If m ≤ n then m-furcus ⇒ n-furcus. Theorem If P is finite and HP = ∅, then HP is m-furcus for some m ≤ |P| + 1. Theorem For every m ≥ 2, there exists a ∈ H with min L(a) ≥ m. Conjecture For every m ≥ 2, there exists a finite P such that HP = ∅ and HP is m-furcus but not (m − 1)-furcus.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

What kind of semigroup is H? H is a finite factorization semigroup H is not root closed

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

What kind of semigroup is H? H is a finite factorization semigroup H is not root closed H is not a Krull monoid

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

What kind of semigroup is H? H is a finite factorization semigroup H is not root closed H is not a Krull monoid H is not a C-monoid

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

What kind of semigroup is H? H is a finite factorization semigroup H is not root closed H is not a Krull monoid H is not a C-monoid However, each HP is a C0-monoid

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Similar monoids? A and D are s-ideals of N≥1 = F(P), a free abelian monoid. To get the multifurcus structure, we need: H is an s-ideal of a free abelian monoid F = F(P) H has no prime powers (i.e. pk / ∈ H for all p ∈ P and k ∈ N)

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

A = {a ∈ N | a is abundant} = {a ∈ N | σ(a) > 2a} D = {a ∈ N | a is nondeficient} = {a ∈ N | σ(a) ≥ 2a} Set f (a) = σ(a)/a, which is a multiplicative function. Then A = {a ∈ N | f (a) > 2} D = {a ∈ N | f (a) ≥ 2} We can generalize to higher levels of abundance: for any real c ≥ 2, we have: H = {a ∈ N | f (a) > c} H′ = {a ∈ N | f (a) ≥ c} are both multifurcus semigroups.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

We can use other multiplicative functions: Example H = {a ∈ N | a is not a prime power} = {a ∈ N | ω(a) ≥ 2} is also multifurcus.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

We can use other multiplicative functions: Example H = {a ∈ N | a is not a prime power} = {a ∈ N | ω(a) ≥ 2} is also multifurcus. Example For any real c ≤ 1/2, H = {a ∈ N | φ(a) < ca} is also multifurcus.

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Question: For which multiplicative functions f : N → R≥0 and which real numbers c, are any of the sets multifurcus semigroups? Hc,> = {a ∈ N | f (a) > c} Hc,≥ = {a ∈ N | f (a) ≥ c} Hc,< = {a ∈ N | f (a) < c} Hc,≤ = {a ∈ N | f (a) ≤ c} The same theory works with additive fuctions f : Ns → R≥0 and additive semigroup ideals of Ns defined using f .

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers

Thank you!

Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers