n-Factorizations in Z€¦ · ˝n-Factorizations in Z Preliminaries Mod n Recall the de nition of...

τn-Factorizations in Z

Nate BishopSt. Olaf College

September 29, 2011

Nate Bishop St. Olaf College τn-Factorizations in Z

τn-Factorizations in ZPreliminaries

Factorization

Usual Factorizations

(1)(7) or (−1)(−7),

6 = (1)(6), (−1)(−6), (1)(2)(3), (1)(−2)(−3), (−1)(−2)(3),or (−1)(2)(−3),

Factorization

7 = (1)(7) or (−1)(−7),

6 = (1)(6), (−1)(−6), (1)(2)(3), (1)(−2)(−3), (−1)(−2)(3),or (−1)(2)(−3),

Factorization

7 = (1)(7) or (−1)(−7),

(1)(6), (−1)(−6), (1)(2)(3), (1)(−2)(−3), (−1)(−2)(3),or (−1)(2)(−3),

Factorization

7 = (1)(7) or (−1)(−7),

6 = (1)(6), (−1)(−6), (1)(2)(3), (1)(−2)(−3), (−1)(−2)(3),or (−1)(2)(−3),

Recall the definition of modular arithmetic:

Definition

If a is congruent to b mod n, then we can say a− b = nr forsome integer r .

Alternatively, if a ≡ b mod n, then a and b have the sameremainder after being divided by n.

Recall the definition of modular arithmetic:

Definition

If a is congruent to b mod n, then we can say a− b = nr forsome integer r .

Alternatively, if a ≡ b mod n, then a and b have the sameremainder after being divided by n.

τn-Factorizations in Zτn

So, what does τn mean?

Definition

Two integers are τn-equivalent if they are congruent mod n.

Example

Let n = 2.

Note that 3 ≡ 1 mod 2.

Then we say that 3 τ2 1.

Definition

Example

Let n = 2.

Definition

Example

Let n = 2.

Definition

Example

Let n = 2.

Definition

Example

Let n = 2.

Definition

Example

Let n = 3.

Is 7τ315?

No, but 7τ316.

Definition

Example

Let n = 3.

Is 7τ315?

No, but 7τ316.

Definition

Example

Let n = 3.

Is 7τ315?

No, but 7τ316.

Definition

A τn-factorization of a ∈ Z \ {−1, 0, 1} is as follows:

a = (±1)(a1 · · · ak),

such that ai τn aj for all i < j ≤ k .

Examples of τ2-Factorizations

Valid τ2-Factorizations:

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

Invalid τ2-Factorizations:

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

7 = 1 · 7, (−1) · (−7)

6 = 1 · 6, (−1) · (−6)

12 = 1 · 12, (−1) · (−12), 2 · 6, (−1) · (−2) · 6, (−1) · 2 · (−6),(−2) · (−6)

6 = 1 · 2 · 3, (−1) · (−2) · 3, (−1) · 2 · (−3)

12 = 3 · 4, (−1) · (−3) · 4, (−1) · 3 · (−4), (−3) · (−4)

How does the usual factorization work into this theory?

Consider τ1-factorization.

Definition

If a is congruent to b mod 1, then we can say a− b = 1 · r forsome integer r .

Definition

A τn-atom is an integer (different from -1,0,1) which has only thetrivial τn-factorizations.

Examples

Primes

If n = 7, then 6 is a τ7-atom.

Definition

Examples

Primes

Definition

Examples

Primes

Definition

Examples

Primes

UFD HFD

BFD ACCP Atomic

UFD HFD

BFD ACCP Atomic

τn-Factorizations in ZAtomic Domains

Definition

Z is a τn-atomic domain if every integer can be τn-factored intoτn-atoms.

Q: For which n ∈ N is Z a τn-atomic domain?

Atomic for 0 ≤ n ≤ 6, n = 8, 10.

Non-atomic for all other natural numbers.

Atomic for 0 ≤ n ≤ 6, n = 8, 10.

Non-Atomic Domains

What does it mean for Z to not be τn-atomic?

There exists an integer whose τn-factorizations are not intoτn-atoms.

Non-Atomic Domains

What does it mean for Z to not be τn-atomic?

There exists an integer whose τn-factorizations are not intoτn-atoms.

Counter Example

Let n=7.

Choose z = 12.

z has many usual factorizations, but it’s only non-trivialτ7-factorizations are 12 = (−1)(−4)(3) or 12 = (−1)(4)(−3);but neither of these factorizations are a factorization of z intoτ7-atoms.

Counter Example

Let n=7.

Choose z = 12.

Counter Example

Let n=7.

Choose z = 12.

Counter Example

Let n=7.

Choose z = 12.

UFD HFD

BFD ACCP Atomic

Definition

Z is a τn-Half Factorial Domain (HFD) if Z is atomic and, givenx ∈ Z and two τn-factorizations ofx = (±1)(a1 . . . an) = (±1)(b1 . . . bm), n = m.

Example

Let n = 2 and z = 72 = 23 · 32. Then two possibleτ2-factorizations of z are 6 · 6 · 2 = 2 · 2 · 18.

Research on τn-HFDs

Z is a τn-HFD for all n ∈ {0, 1, 2, . . . , 6, 8, 10}.(Conjecture) Z is not a τn-HFD with completeness for allother values of n.

Z is a τn-HFD for all n ∈ {0, 1, 2, . . . , 6, 8, 10}.

(Conjecture) Z is not a τn-HFD with completeness for allother values of n.

Z is a τn-HFD for all n ∈ {0, 1, 2, . . . , 6, 8, 10}.(Conjecture) Z is not a τn-HFD with completeness for allother values of n.

Conclusion

τn-factorization

τn-atoms

τn-atomic domains

τn-HFDs

Future Work

Conclusion

τn-factorization

τn-atoms

τn-atomic domains

τn-HFDs

Future Work

Conclusion

τn-factorization

τn-atoms

τn-atomic domains

τn-HFDs

Future Work

Conclusion

τn-factorization

τn-atoms

τn-atomic domains

τn-HFDs

Future Work

Conclusion

τn-factorization

τn-atoms

τn-atomic domains

τn-HFDs

Future Work

Conclusion

τn-factorization

τn-atoms

τn-atomic domains

τn-HFDs

Future Work

Thanks to:Dr. Dan Anderson,The University of Iowa Department of Mathematics NSF VIGREgrant No. 0602242 and Alliance grant No. 0502354.

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