[seqfan] Re: Sequences counting distinct factorizations in a non-UFD

[David Wilson]

In a ring of quadratic integers Z[sqrt(n)], the norm of element is N(a + b * sqrt(n)) = |a^2 - b^2 *n|.
A unit u in Z[sqrt(n)] is an element with N(u) = 1.
An element p of Z[sqrt(n)] is irreducible if p = a * b implies a or b is a unit.
Two irreducibles p and q are equivalent if there exists unit u with q = p * u.
If two irreducibles p and q are equivalent, then N(p) = N(q).
Two factorizations of a = p1*p2*...*pk = q1*q2*...*qk are equivalent if the factors can be ordered such that each p_i is equivalent to q_i.

In Z[sqrt(10)], assume 2, 3, 4+sqrt(10), and 4-sqrt(10) are irreducible.
Then the factors in the factorization 36  = 2^2 * 3^2 have norms (4, 4, 9, 9).
In the factorization 2 * 3 * (4+sqrt(10)) * (4-sqrt(10)), they are (4, 9, 6, 6).
In the factorization (4+sqrt(10))^2 * (4-sqrt(10))^2, they are (6, 6, 6, 6).

Since the norms cannot be ordered to match between any of the two factorizations, all three are distinct.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Alonso
> Del Arte
> Sent: Tuesday, October 28, 2014 9:46 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Sequences counting distinct factorizations in a non-UFD
> 
> Should've written "2 * 3 * (4 - sqrt(10))(4 + sqrt(10)) count as a distinct
> factorization apart from 2^2 * 3^2 and (4 - sqrt(10))^2 (4 + sqrt(10))^2"
> in  that previous message. Sorry.
> 
> On Tue, Oct 28, 2014 at 9:44 PM, Alonso Del Arte
> <alonso.delarte at gmail.com>
> wrote:
> 
> > I suspect some sequences like this one, if not exactly this one, are
> > already in the OEIS, but I haven't had the time to sit down to work out the
> > theory and verify the identifications:
> >
> > a(n) counts how many distinct factorizations n has in Z[sqrt(10)].
> >
> > It is a very well-known fact that a(6) = 2.
> >
> > But I haven't seen discussion anywhere of what a(36) is. Does 6 * (4 -
> > sqrt(10))(4 + sqrt(10)) count as a distinct factorization apart from 6^2
> > and (4 - sqrt(10))^2 (4 + sqrt(10))^2? Should it count?
> >
> > Al
> >
> > P.S. I am aware of the all 1's sequence being the equivalent for UFDs.
> >
> > --
> > Alonso del Arte
> > Author at SmashWords.com
> > <https://www.smashwords.com/profile/view/AlonsoDelarte>
> > Musician at ReverbNation.com
> <http://www.reverbnation.com/alonsodelarte>
> >
> 
> 
> 
> --
> Alonso del Arte
> Author at SmashWords.com
> <https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com
> <http://www.reverbnation.com/alonsodelarte>
> 
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