Sunday, August 27, 2006

 

Random prime behavior vs politics

Supposedly there is a debate about how random prime numbers are, but recently I noticed that you could do this simple thing, of considering p mod 3, where p is a prime greater than 3.

So what is p mod 3? It's like clock-time, where when you get to 12, you start over with 1, so for instance 7 mod 3 = 1, and 15 mod 3 = 0, so another way to think about it is, subtract 3 until doing so would give you a negative number, and what's left over is what is called the residue modulo 3.

The residue modulo 3 of 7 is 1. 7 mod 3 = 1.

So it's a simple idea, but if you consider the first 23 primes greater than 3 in order, and look at p mod 3, you get a series:

2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1

And you can check that easily enough as 5 mod 3 = 2, which is why it starts with 2, and as I said above 7 mod 3 = 1, so next is 1 and so forth.

That series is random. There is no rhyme nor reason to it, and given the series, you cannot predict what the next number will be, except to say it will be 1 or 2.

If 1 were heads and 2 tails that could be a series of coin flips.

And why wouldn't it be random? Why should it matter to a prime like 5 what its residue modulo 3 is? Or to 7? Or to greater primes?

You can do the same thing with other primes, and here is p mod 5, with p greater than 5:

2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2

If you think there is a pattern I suggest you extend it out further. It is a random list.

In general p_1 mod p_2 with p_1 greater than p_2, where both are primes will give you a random list, because in every case the larger primes don't care what their residue modulo the smaller prime is, so you get this easy answer.

But there is a debate within the mathematical community about whether or not primes are random.

So what gives?

Well, consider twin primes. By the standard terminology you have twin primes when one prime is 2 away from another, like 5 and 7 are twin primes, as are 11 and 13.

So you can write twin primes as

p_1 - p_2 = 2

so

p_1 mod p_2 = 2

and I've gone over how primes are random in this area.

That indicates that being twin prime is random and is just about probability, and nothing else.

But if you do a web search on "twin primes" you will read about a lot of research in the area.

How can people do a lot of research on something that is random?

Well, they can if it's not known to be random!!!

You see, mathematicians doing research in this area are doing nothing at all, as the twin primes are random, so there is no reason to them beyond probabilities, like with p mod 3, the probability that it is 1 is 50% as is the probability it is 2.

There is no further information available as the primes don't care about their residue modulo a lesser prime.

With that information, you can get the best answer possible about twin primes—which is simple but I won't go into detail to keep this a simple post—and there is then NO FURTHER INFORMATION POSSIBLE.

So the mathematics in this area should be a couple of paragraphs in some beginning number theory textbook.

But do that web search on "twin primes" and see how much has been written as if there is the possibility of learning more—about something random.

Why would people do such a thing?

Do you have a job? What if your job were being brilliant?

What if to keep your job and live up to the expectations of people around you, you needed to keep being brilliant—or look like you were brilliant—year after year?

Heard of publish or perish?

What better way to keep a career going than to do research on a random process that people don't realize is random?

You can work your entire career—looking brilliant, if they don't know the trick—doing nothing at all.





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