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 d | a Each a ≥ 2 is abundant if σ ( a ) > 2 a perfect if σ ( a ) = 2 a deficient if σ ( a ) < 2 a A = { a ∈ N | a is abundant } = { a ∈ N | σ ( a ) > 2 a } = { 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 ) ≥ 2 a } = { 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: A P = { a ∈ A | Supp ( a ) ⊆ P } D P = { a ∈ D | Supp ( a ) ⊆ P } Example: A { 3 , 5 , 7 } = { a = 3 e 1 5 e 2 7 e 3 | a is abundant } Facts: 1 If | P | = 1, then A P = D P = ∅ . 2 If | P | = 2 and 2 / ∈ P , then A P = D P = ∅ . 3 As min P → ∞ , we must have | P | → ∞ to guarantee A P � = ∅ (and D P � = ∅ ). 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 b σ ( 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 b σ ( 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 = { p 1 , . . . , p s } ⊆ P finite, if H P � = ∅ , then H P is a divisor-closed subsemigroup of H and H is isomorphic to an additive subsemigroup of N s . Dickson’s Theorem says H P has only finitely many primitives. So does H P 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 = { p 1 , . . . , p s } ⊆ P finite, if H P � = ∅ , then H P is a divisor-closed subsemigroup of H and H is isomorphic to an additive subsemigroup of N s . Theorem For any P ⊆ P finite, if H P � = ∅ , then H P 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 · 120 945 3 = 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 H P ⊆ qr N for some distinct primes q , r, then H P is bifurcus. Example Let P = { 3 , 5 , 7 , 11 } . Every nondeficient number with support in P is divisible by 15. Since H P ⊆ 15 N , so H P 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 H P ⊆ qr N for some distinct primes q , r, then H P is bifurcus. Example Let P = { 3 , 5 , 7 , 11 } . Every nondeficient number with support in P is divisible by 15. Since H P ⊆ 15 N , so H P is bifurcus. However, for P = { 3 , 5 , 7 , 11 , 13 } , H P �⊆ qr N for any q , r ∈ { 3 , 5 , 7 , 11 , 13 } , because a = 3 3 · 5 · 7, b = 3 3 · 5 2 · 11, and c = 3 2 · 7 2 · 11 2 · 13 3 are elements of H P . So the theorem does not apply to this H P . 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 H P � = ∅ , then H P 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 H P � = ∅ , then H P 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 H P � = ∅ and H P 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 H P � = ∅ , then H P 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 H P � = ∅ and H P 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 H P is a C 0 -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. p k / ∈ 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 ) > 2 a } D = { a ∈ N | a is nondeficient } = { a ∈ N | σ ( a ) ≥ 2 a } 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? H c ,> = { a ∈ N | f ( a ) > c } H c , ≥ = { a ∈ N | f ( a ) ≥ c } H c ,< = { a ∈ N | f ( a ) < c } H c , ≤ = { a ∈ N | f ( a ) ≤ c } The same theory works with additive fuctions f : N s → R ≥ 0 and additive semigroup ideals of N s defined using f . Paul Baginski Fairfield University Nonunique Factorization of Abundant Numbers
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