irregular primes

[arguchik]

at a well-attended brunch on saturday, i asked everyone at the table what an irregular prime number is (i know that the first irregular prime is 37, but i don't know why). here's a link to the wiki article on the subject. i can't figure out what the fuck they're talking about. can anyone out there explain this to me in plain english?

some context: i asked about this, because i am teaching an excerpt from ralph ellison's invisible man at the moment--specifically the prologue and the first chapter (which is often anthologized as a stand-alone short story called "battle royal"). in the prologue, the narrator mentions twice that he has exactly 1369 lightbulbs in the "hole" he lives in. i was curious about the number, so i did some quick factoring and figured out that it is the square of 37; and that its only factors are 1, 37, and itself. i wiki'ed the number 37, and that's where i found out that it's the first irregular prime.

Comments

[rojonoir.livejournal.com]

Wow.

"In mathematics, a regular prime is a certain kind of prime number. A prime number p is regular if it does not divide the class number of the p-th cyclotomic field (that is, the algebraic number field obtained by adjoining the p-th root of unity to the rational numbers)."

All you need to know is what these are:

• prime number
  
• does not divide
  
• class number
  
• p-th cyclotomic field
  
• algebraic number field
  
• p-th root of unity
  
• rational numbers
  
• adjoining the p-th root of unity to the rational numbers
  

A prime number can only be divided by itself and 1. A number doesn't divide if you get a remainder - 2 divides 4, but 2 doesn't divide 5. Pth root of unity is the square root of 1 if P=2, the cube root of 1 if P=3, etc. So the 37th root of unity is 1^(1/37). A rational number is a number that can be expressed as a fraction n/d where both n and d are whole numbers.

Not sure how one adjoins the pth root of unity to the rational numbers, or what a class number is, though.

Most of those wikipedia math pages seem to be aimed at people already in the specific field, as even after years of Computer Science grad school (with some math), it mostly looks like gibberish to me. As Barbie would say, "Math is hard."

[arguchik.livejournal.com]

i am barbie. hear me roar.

[glaucon.livejournal.com]

you could try emailing this guy.

he's an old friend of mine (with whom I haven't spoken in about a decade) and blogs a lot about math. he's also one of the three smartest people I've ever met.

caveat

[glaucon.livejournal.com]

there's at least a 50% chance that his response, if any, will be "go to the fucking library".

[arguchik.livejournal.com]

thank you--though in light of your subsequent post and ursula's resonse to my post, i'll probably leave him be for now.

bear with me, it's been a while

[alice-at-night.livejournal.com]

Here's a constructive approach
A. these things called "Bernoulli numbers" exist and their values can be calculated.

B. Bernoulli numbers have a numerator (and a denominator, but the denominator isn't important in this discussion).

C. Someone's done the math: on this page the second column labeled a(n) is a list of the first 40 numerators . http://www.research.att.com/~njas/sequences/table?a=27641&fmt=4

D. If the prime can evenly divide one of these numerators evenly, it is an "irregular prime"

E. 37 divides the numerator of B(32) = - 7709321041217 evenly, so it is imperfect. 691 is also imperfect.

There is some special rule that if the prime has to be bigger than the Bernoulli number or it doesn't count. Which is why 5 & 7 don't count. I.e., 5 is not bigger than its B-number of 10; 7 isn't bigger than its B-number of 14 (Actually the rule is: bigger than 2 time the B value plus one, but that is getting a bit detailed)

The above steps will find some but not all imperfect primes - since Bernoulli numbers aren't the only "p-th cyclotomic field" type numbers.

I'm about 89% sure the above is correct. It's been a long long time.

Re: bear with me, it's been a while

[arguchik.livejournal.com]

wow! thank you!

i'm trying to think, now, about why ellison might have been drawing on this mathematical stuff in invisible man, and what it might mean in the context of the novel. ellison studied music at the tuskegee institute--he was going to be a composer, and the novel deals a lot with jazz. i wonder if the bernoulli nmbers have some musical application... i just don't have any information as to whether or not ellison had mathematical training, so this whole thing might just be a coincidence. he does a lot of stuff in the novel with the divisibility, multiplicity, and interpenetrations of blackness and whiteness, too.