The Lumber Room

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Archive for September 5th, 2009

Testing irreducibility using prime numbers

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Here’s a simple and nice test for irreducibility in {\mathbb{Z}[x]} that N told me about a year ago. (I just noticed this lying around while cleaning; I don’t have a year’s buffer like Raymond Chen.) Apologies for the ugly formatting; you’ll have to trust that the result (Theorem 1, or Corollary 8) is more beautiful than it looks. :-)

Actually I’m not sure why I wrote this originally, given that it’s all already well-explained in the originals and even partially on Wikipedia. Perhaps my proofs are different or simpler or I was bored or something.

1. Irreducibility test

In its simplest form, the test can be stated as follows.

Theorem 1 Given a polynomial {f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_o} with integer coefficients, let {G = \max_{i}|\frac{a_i}{a_n}|}. If there exists an integer {m \ge G+2} such that {f(m)} is prime, then {f} is irreducible.

For example, with the polynomial x^2 + 3x + 1, we have G = 3, and f(5)=f(G+2)=41 is prime, which proves that it is irreducible. (We could also evaluate f at e.g. 7, 8, 9, or 10 to get the same conclusion.)

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Written by S

Sat, 2009-09-05 at 22:01:44

Conscious consumption

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Late-night sleepy ramblings; please do not read :p

I have been taking a break, and it has helped me gain some perspective. Or so I thought.

Like some who might be reading this, I subscribe to a large number of blogs. Google Reader says 106 subscriptions, but a few of them are aggregators which combine the updates from several blogs.

For about three months (since June 10th, I think), I have not been reading them, nor reading the news. I’m not exactly sure why… it started as a day’s break (which was a big deal), then became four days (which was an even bigger deal), then it got easier and easier. Probably, I thought I was taking a break from (parts of) the internet in order to catch up with (parts of) my life. It didn’t work, of course. I merely found other sinks in which to dump my time. (I spent more time on Wikipedia than ever before, read more actual books than I had in the last couple of years, and so on.)

I did, however, discover a couple of things.

One is that Google Reader stops updating the count of unread items at “1000+”. (It also automatically marks items more than 30 days old as read, and, as I have “only” about 1500 items a month, I don’t know if it counts to “2000+”.)

The other is some general observations about what our lives have become.

It seems that the meanings of words like “recreation” have become somewhat quaint. Now “entertainment” is not always something to indulge in because one requires relaxation, or because it is a rewarding pursuit in itself, but simply “because it’s there”.
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Written by S

Sat, 2009-09-05 at 02:49:44

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