Prime Numbers Not So Random?

anyone else getting the feeling... by ddd2k · 2003-03-24 20:08 · Score: 4, Interesting

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.

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Great Atrocit

Re:anyone else getting the feeling... by Yokaze · 2003-03-24 21:29 · Score: 3, Insightful

> In the scientific community, proof is established by repeated experimental repetition, in Mathematics, testing this theory lots of times with lots of different numbers (see computers)

Sorry. Say that to a mathematician, and see how he laughs at you, or kicks you out.

A theorem is statement which can be verified by mathematical operations.

The statements usually includes axioms, which are not provable, and which define the mathematical operations on the given problem. The only thing you have to do is show is, given these axioms, the statement is always true.

The sequence of application of axioms is called "proof".

And mathematicians are very peculiar with "always". For scientists "always" means "many times, and until some shows otherwise", because you can't define these axioms.

Mathematics is not the kind of science you are thinking of. You are thinking of natural science.
In mathematics, humans define the axioms. They may, or may not bear any relationship to reality.
It some aspects, it resembles more philosophy than physics.

> Who are you to say that the editors of Nature don't know what to publish?

Probably a mathematician. They don't give a lot on physicists saying, "Hey, by finding some statistical correlation, we found a pattern, which holds true, in our finite data set, most of the time".

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"Between strong and weak, between rich and poor [...], it is freedom which oppresses and the law which sets free"
Re:anyone else getting the feeling... by Scarblac · 2003-03-24 21:36 · Score: 3, Informative

You've "easily" proven things by defining them as something. An irrational number is a number with no known, infinite, repeatable sequence? You've *defined* it that way, that doesn't mean you've ever *proven* a number irrational.

Proof that the square root of 2 is irrational: http://everything2.com/index.pl?node_id=928307

Proof that e is irrational: http://everything2.com/index.pl?node_id=930313

Better examples are obviously out there, but I just searched for 'irrational' on E2... You're very ignorant.

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Re:anyone else getting the feeling... by Smidge204 · 2003-03-25 00:27 · Score: 4, Funny

Proof: All odd numbers are prime.

Mathematitian: "1 is prime, 3 is prime, 5 is prime, 7 is prime. The rest are prime by induction."

Physisist: "1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is not but is likely to be experimental error, 11 is prime, 13 is prime..."

Engineer: ""1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime to a reasonable degree of accuracy, 11 is prime, 13 is prime..."

Computer Scientist: "1 is prime, 1 is prime, 1 is prime, 1 is prime..."

</joke>
=Smidge=

Encryption? by asdfx · 2003-03-24 20:15 · Score: 3, Informative

I wonder if this theory could be used to produce code that could be useful for encryption based on prime numbers, such as RSA's work. Would it make it easier to produce reliable prime numbers much larger than 1024 or even 2048 bit? Further, I wonder if this could be used to drastically reduce the time required to brute force an RSA encrypted message. Could the encryption of files that were encrypted with 128 bit technology be rendered all but useless?

figure & ground by obtuse · 2003-03-24 20:46 · Score: 3, Interesting

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.

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Assembly is the reverse of disassembly.

Here's the rub by Anonymous Coward · 2003-03-24 21:20 · Score: 5, Interesting

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.

Re:Here's the rub by Anonymous Coward · 2003-03-24 21:37 · Score: 3, Informative

They don't claim that they have a rule that can create prime numbers, they just claim that prime numbers might not be completely random.

Just like if you have a large prime p, p+210 is 4.375 times more likely to be a prime than a random integer around p. Not a rule, but a hint that primes aren't so random.

Re:I found a pattern! by itsme1234 · 2003-03-25 01:07 · Score: 5, Insightful

Of course they are prime ! ANY number is either:

6n (not prime of course)
6n+1
6n+2 (not prime of course)
6n+3 (not prime of course)
6n+4 (not prime of course)
6n+5

And 6n+5 is the same as 6(n+1)-1 so indeed you are right. You deserve a price for finding a 6th grade theorem.

Re:Physicists pulling a cold fusion? by l2718 · 2003-03-25 05:54 · Score: 3, Informative

As a number theory graduate student, this looks suspicious. This isn't as bad as last summer, when some string theorists claimed a junk
proof of the Riemann Hypothesis, but it's close.

Prime numbers are very hard to tackle. Part of the difficulty in this style of problem, as another post points out, is that they are defined multiplicatively, and yet we here care about additive properties (differences in this case).

I have a few concerns with this paper:

1. They look at a really small number of primes (only 10^7 of them). Many false conjectures have been made that way. The most famous case is with the prime number theorem: it's known that up to x there are about x/log(x) primes, and as x grows this estimate becomes more and more accurate. If you do some tests you'll quickly see that there are more than x/log(x) primes up to x for all x you can test for. This was conjectured to be true for all x, until someone proved that actually the difference (# primes up to x) - x/log(x) changes sign infinitely often. The first change is known to happen before x=10^370 -- but try testing that.

2. They use the ansatz Alog(log(x))+B to fit some function of x (the entropy). But for x in the range of concern (at most 10^8), log(log(x)) is essentially constant. Try graphing that function and you'll see for yourself. For all practical purposes (i.e. unless you can run your computer up to numbers like 10^100), doing curve fitting with this function is very suspicious.

My take,
Lior

How to prove that all odd numbers are prime by dargaud · 2003-03-25 08:02 · Score: 4, Funny

"It was mentioned on CNN that the new prime number discovered recently is four times bigger than the previous record." John Blasik

"You know what seems odd to me? Numbers that aren't divisible by two." Michael Wolf.

"I don't get even, I get odder."

Well, the problem "How to prove that all odd numbers are prime" has different solutions whether you are a:

Mathematician: 1 is prime, 3 is prime, 5 is prime, 7 is prime, and by induction we have that all the odd integers are prime.

Physicist: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error...

Engineer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime...

Chemist: 1 prime, 3 prime, 5 prime... hey, let's publish!

Modern physicist using renormalization: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is ... 9/3 is prime, 11 is prime, 13 is prime, 15 is ... 15/3 is prime, 17 is prime, 19 is prime, 21 is ... 21/3 is prime...

Quantum Physicist: All numbers are equally prime and non-prime until observed.

Professor: 1 is prime, 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student.

Confused Undergraduate: Let p be any prime number larger than 2. Then p is not divisible by 2, so p is odd. QED

Measure nontheorist: There are exactly as many odd numbers as primes (Euclid, Cantor), and exactly one even prime (namely 2), so there must be exactly one odd nonprime (namely 1).

Cosmologist: 1 is prime, yes it is true....

Computer Scientist: 1 is prime, 10 is prime, 11 is prime, 101 is prime...

Programmer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 will be fixed in the next release, ...

C programmer: 01 is prime, 03 is prime, 05 is prime, 07 is prime, 09 is really 011 which everyone knows is prime, ...

BASIC programmer: What's a prime?

COBOL programmer: What's an odd number?

Windows programmer: 1 is prime. Wait...

Mac programmer: Now why would anyone want to know about that? That's not user friendly. You don't worry about it, we'll take care of it for you.

Bill Gates: 1. No one will ever need any more than 1.

ZX-81 Computer Programmer: 1 is prime, 3 is prime, Out of Memory.

Pentium owner: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 8.9999978 is prime...

GNU programmer: % prime
usage: prime [-nV] [--quiet] [--silent] [--version] [-e script] --catenate --concatenate | c --create | d --diff --compare | r --append | t --list | u --update | x -extract --get [ --atime-preserve ] [ -b, --block-size N ] [ -B, --read-full-blocks ] [ -C, --directory DIR ] [--checkpoint ] [ -f, --file [HOSTNAME:]F ] [ --force-local ] [ -F, --info-script F --new-volume-script F ] [-G, --incremental ] [ -g, --listed-incremental F ] [ -h, --dereference ] [ -i, --ignore-zeros ] [ --ignore-failed-read ] [ -k, --keep-old-files ] [ -K, --starting-file F ] [ -l, --one-file-system ] [ -L, --tape-length N ] [ -m, --modification-time ] [ -M, --multi-volume ] [ -N, --after-date DATE, --newer DATE ] [ -o, --old-archive, --portability ] [ -O, --to-stdout ] [ -p, --same-permissions, --preserve-permissions ] [ -P, --absolute-paths ] [ --preserve ] [ -R, --record-number ] [ [-f script-file] [--expression=script] [--file=script-file] [file...]
prime: you must specify exactly one of the r, c, t, x, or d options
For more information, type "prime --help''

Unix programmer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, ...
Segmentation fault, Core dumped.

Computer programmer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 9 is prime, 9 is prime, 9 is ...

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