Prime Numbers Not So Random?

Jeff Moriarty writes "Some physicists believe they might have caught a whiff of a pattern in the sequence of prime numbers. This would have a huge impact across mathematics, and to people who just really like primes... or like being Prime."

Comments

anyone else getting the feeling...

[ddd2k]

the interval thing seemed like such a trivial observation. surely many others have easily noticed that. Its another "I think i discovered a pattern" claim, while still have no way to prove it.

Re:anyone else getting the feeling...

[WatertonMan]

While traditionally there has been a wide gap between physics and mathematics in terms of method, quite a few philosophers have argued that this gap has been closing. One big reason for it is the advent of the computer. Many of the most famous mathematical discoveries of the last decade have involved a *lot* of computer work. To such an extent that suggesting that anyone has actually gone through every step of the proof is likely wrong. Further studies in fields like chaos have tended to use computers like a laboratory, utilizing a lot of the methadology of physics.

One good paper on this phenomena of how proof has changed in math is William Thurston's, "On Proof and Progress in Mathematics." A great book that goes through a lot of recent work in philosophy of mathematics is New Directions in the Philosophy of Mathematics

by the Princeton University Press. While I still am primarily a Constructivist, I find that recent anti-foundationalist approaches to mathematics are very interesting. In particular you might find Putnam's assertion that math is becoming quasi-empirical very interesting.

I always find Putnam interesting, even though I often disagree with him. He tends to try to blur the realist - anti-realist distinction in ways that end up being more slight of hand. (IMO)

Anyway, I think that you are adopting a foundational view of mathematics for which there are good reasons to reject. Further we can have a good reason for thinking a theorem correct without necessarily being able to prove it from strict foundations. And indeed the crisis in mathematical foundations at the end of the 19th century was the recognition that math had been proceeding without doing what you suggest. That was partially the impetitus behind such things as Whitehead and Russell's work on mathematical foundations.

figure & ground

[obtuse]

Yeah, I remember being excited when I saw a graph of primes that were dots in a field of blank composites. There were lines & patterns all over the place. Wow!

Then I realized that the composite numbers will each make a pattern in any graph. By their nature they repeat.

What I was looking at was the space in between the patterns created by the composites. For example, all primes are odd. There's a set of straight lines on any graph. Well, it's more enlightening to say that none are even, becasue then they'd be divisible by two. Each new set of composites creates another pattern that makes a hole in possible primes.

Here's the rub

[Anonymous Coward]

Here's the problem with finding patterns in Primes: It has to do with the way most things in number theory are formulated. Prime numbers are figured out by a process of non-definition and NOT by some form of additive process. An example or two might make that statement a bit clearer:

If I needed, for example, to find a rule that returns only even numbers, my problem is simplicity itself, I have no need to test a given number to determine whether or not it is even, I can force it to be even by applying any number of simple (or complex) formulas that work within the system.

If someone gives me number X, I have no need to know what X is, all I have to do is multiply X by 2 and (after a little inductive reasoning), I have guaranteed that I now have an even number.

Prime numbers are NOT found that way. An even number is determined to have the property 'evenness' from within the number system itself, namely multiplication by 2. It is a simple additive process to include other even numbers into a given set. A prime number on the other hand, forgive the inexactness, can be considered to have the inherent property 'whatever property that created me that is unique to me'.

IOW, each prime number is unalterably unique and furthermore it is unique in a way which is unique to EACH AND EVERY prime number, all by itself. No other prime number has the same property that makes any other prime number unique.

EXAMPLES (bad, I know, but the best I could do at 0430):
the number 7 (a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 7 and 1, a property that it shares with no other numbers.

the number 17 (a prime) has the unique property (among other properties, like 'oddness') that it has the unique divisors 17 and 1, a property that it shares with no other numbers.

the number 21 (not a prime) has the property (among other properties, like 'oddness') that it has the divisors (7 and 3) AND (21 and 1). Only primes get to leave out that AND part.

The prime numbers are the GAPS within the number-system (and in a rather pathological side note - they are also the glue that holds the system together). The definition of a prime number is, put simplistically: ANY number X that is NOT composite.

Saying you have found a pattern in the prime numbers is tantamount to saying that you have a rule that can create prime numbers W/O checking to see if it's true or not. Put another way, it is exactly the same as saying:

"I have a formula P(x) that can always churn out primes, give me a number, any number and after the application of my formula, I can guarantee that it will be a prime number."

If you could do that, I have a whole bunch of NP complete problems for you to work on (and a bone to pick with a certain Mr. Godel).

Any pattern w/in the set of prime numbers would be a formula with an infinite number of rules (an individual rule for each individual prime number, AT LEAST), and anything with an infinite number of rules can be considered completely, totally and irrevocably RANDOM.

Some late night ramblings from a guy who's too tired and lazy to log on.

Anyone can test this theory out.

[pauldy]

Take the first 1000 primes from the site listed. Put them in your favorite spreadsheet. Then use the formula they give to find out they are mostly full of it. For they first few it looks like a pattern is forming then it looks like nothing but noise when plotted. I can't believe no one even tryed this before they actually published this article.

Parallels with Carl Sagan's "Contact"

[Anonymous Coward]

This sounds spookily like the ending in Contact (only in the book, not the film) where researchers find a message buried in the seemingly random digits of Pi. The implication was that the builders of the universe had left behind their signature.

Perhaps these guys should map out their sequences of prime number differences to see if it generates a picture ?

Some numbers to look at

[big_alpaca]

So here is 5 minutes of mathemetica output to look at. dlist is the list of differences between sequential primes. ilist is the increment between subsequent differences. Have fun.

ilist = dlist = {};
Do[dlist = Append[dlist, Prime[n + 1] - Prime[n]], {n, 1000}];
Do[ilist = Append[ilist, dlist[[n + 1]] - dlist[[n]]], {n, 999} ];

dlist
{1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 6, 8, 18, 10, 14, 4, 2, 4, 6, 8, 4, 2, 6, 12, 10, 2, 4, 2, 4, 6, 12, 12, 8, 12, 6, 4, 6, 8, 4, 8, 4, 14, 4, 6, 2, 4, 6, 2, 6, 10, 20, 6, 4, 2, 24, 4, 2, 10, 12, 2, 10, 8, 6, 6, 6, 18, 6, 4, 2, 12, 10, 12, 8, 16, 14, 6, 4, 2, 4, 2, 10, 12, 6, 6, 18, 2, 16, 2, 22, 6, 8, 6, 4, 2, 4, 8, 6, 10, 2, 10, 14, 10, 6, 12, 2, 4, 2, 10, 12, 2, 16, 2, 6, 4, 2, 10, 8, 18, 24, 4, 6, 8, 16, 2, 4, 8, 16, 2, 4, 8, 6, 6, 4, 12, 2, 22, 6, 2, 6, 4, 6, 14, 6, 4, 2, 6, 4, 6, 12, 6, 6, 14, 4, 6, 12, 8, 6, 4, 26, 18, 10, 8, 4, 6, 2, 6, 22, 12, 2, 16, 8, 4, 12, 14, 10, 2, 4, 8, 6, 6, 4, 2, 4, 6, 8, 4, 2, 6, 10, 2, 10, 8, 4, 14, 10, 12, 2, 6, 4, 2, 16, 14, 4, 6, 8, 6, 4, 18, 8, 10, 6, 6, 8, 10, 12, 14, 4, 6, 6, 2, 28, 2, 10, 8, 4, 14, 4, 8, 12, 6, 12, 4, 6, 20, 10, 2, 16, 26, 4, 2, 12, 6, 4, 12, 6, 8, 4, 8, 22, 2, 4, 2, 12, 28, 2, 6, 6, 6, 4, 6, 2, 12, 4, 12, 2, 10, 2, 16, 2, 16, 6, 20, 16, 8, 4, 2, 4, 2, 22, 8, 12, 6, 10, 2, 4, 6, 2, 6, 10, 2, 12, 10, 2, 10, 14, 6, 4, 6, 8, 6, 6, 16, 12, 2, 4, 14, 6, 4, 8, 10, 8, 6, 6, 22, 6, 2, 10, 14, 4, 6, 18, 2, 10, 14, 4, 2, 10, 14, 4, 8, 18, 4, 6, 2, 4, 6, 2, 12, 4, 20, 22, 12, 2, 4, 6, 6, 2, 6, 22, 2, 6, 16, 6, 12, 2, 6, 12, 16, 2, 4, 6, 14, 4, 2, 18, 24, 10, 6, 2, 10, 2, 10, 2, 10, 6, 2, 10, 2, 10, 6, 8, 30, 10, 2, 10, 8, 6, 10, 18, 6, 12, 12, 2, 18, 6, 4, 6, 6, 18, 2, 10, 14, 6, 4, 2, 4, 24, 2, 12, 6, 16, 8, 6, 6, 18, 16, 2, 4, 6, 2, 6, 6, 10, 6, 12, 12, 18, 2, 6, 4, 18, 8, 24, 4, 2, 4, 6, 2, 12, 4, 14, 30, 10, 6, 12, 14, 6, 10, 12, 2, 4, 6, 8, 6, 10, 2, 4, 14, 6, 6, 4, 6, 2, 10, 2, 16, 12, 8, 18, 4, 6, 12, 2, 6, 6, 6, 28, 6, 14, 4, 8, 10, 8, 12, 18, 4, 2, 4, 24, 12, 6, 2, 16, 6, 6, 14, 10, 14, 4, 30, 6, 6, 6, 8, 6, 4, 2, 12, 6, 4, 2, 6, 22, 6, 2, 4, 18, 2, 4, 12, 2, 6, 4, 26, 6, 6, 4, 8, 10, 32, 16, 2, 6, 4, 2, 4, 2, 10, 14, 6, 4, 8, 10, 6, 20, 4, 2, 6, 30, 4, 8, 10, 6, 6, 8, 6, 12, 4, 6, 2, 6, 4, 6, 2, 10, 2, 16, 6, 20, 4, 12, 14, 28, 6, 20, 4, 18, 8, 6, 4, 6, 14, 6, 6, 10, 2, 10, 12, 8, 10, 2, 10, 8, 12, 10, 24, 2, 4, 8, 6, 4, 8, 18, 10, 6, 6, 2, 6, 10, 12, 2, 10, 6, 6, 6, 8, 6, 10, 6, 2, 6, 6, 6, 10, 8, 24, 6, 22, 2, 18, 4, 8, 10, 30, 8, 18, 4, 2, 10, 6, 2, 6, 4, 18, 8, 12, 18, 16, 6, 2, 12, 6, 10, 2, 10, 2, 6, 10, 14, 4, 24, 2, 16, 2, 10, 2, 10, 20, 4, 2, 4, 8, 16, 6, 6, 2, 12, 16, 8, 4, 6, 30, 2, 10, 2, 6, 4, 6, 6, 8, 6, 4, 12, 6, 8, 12, 4, 14, 12, 10, 24, 6, 12, 6, 2, 22, 8, 18, 10, 6, 14, 4, 2, 6, 10, 8, 6, 4, 6, 30, 14, 10, 2, 12, 10, 2, 16, 2, 18, 24, 18, 6, 16, 18, 6, 2, 18, 4, 6, 2, 10, 8, 10, 6, 6, 8, 4, 6, 2, 10, 2, 12, 4, 6, 6, 2, 12, 4, 14, 18, 4, 6, 20, 4, 8, 6, 4, 8, 4, 14, 6, 4, 14, 12, 4, 2, 30, 4, 24, 6, 6, 12, 12, 14, 6, 4, 2, 4, 18, 6, 12, 8}

ilist
{1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, -4, -2, 2, -2, 2, 10, -10, 2, -4, 8, -8, 4, 0, -2, 2, 0, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 8, -4, 0, 0, -4, 4, -2, -2, 8, 4, -10, -2, 2, 10, -8, 4, -8, 2, 2, 2, -2, 0, -2, 2, 2, -4, 4, 2, -8, 8, -8, 4, -2, 2, 2, -4, -2, 2, 8, -4, -4, 4, -4, 2, 6, -10, 16, -12, 4, -4, 0, -4, 4, 4, -4, 0