Prime numbers

[tentacle]

Someguy and Peter

 

If you wish to keep referring to finding primes within a range as being able to produce them through a process them then there is nothing we can do to stop you; but be aware that this is not normal usage.

 

The crux of the problem with primes, the reason they are so interesting, why so many careers are spent looking at them is that they are not the result of a function or series; they are not able to be generated, there is no pattern in their occurrence

 

 

i totaly disagree with this kind of statements. because we dont understand formula behind them that doesnt mean thy are random. beyond our chaos universe is order.

 

edit: as numero uno breaktrough in all fields of science we need math breaktrough. because with it we will have physics breaktrough.

imho, solution to prime numbers is in geometry.

Edited September 2, 2014 by tentacle

[Bignose]

i totaly disagree with this kind of statements. because we dont understand formula behind them that doesnt mean thy are random. beyond our chaos universe is order.

You can totally disagree all you want, nonetheless, the simple truth is that with our knowledge as of right now, no pattern has been found. If you disagree, let's see proof of a pattern. No 'intuition', no 'we need a breakthrough', but an actual demonstration of a pattern. Without that, the word random is a completely valid and accurate description of our knowledge today.

[Sensei]

Without that, the word random is a completely valid and accurate description of our knowledge today.

 

Nonsense.

Primes don't fulfill definition of random f.e.

http://www.macmillandictionary.com/us/thesaurus/american/random

"chosen or happening without any particular method, pattern, or purpose"

 

Probability of that 2,3,5,7,11,13 etc etc. is prime number is 100%.

 

Pattern to predict next prime is completely not needed. You don't ask for next even number while examining f.e. 10.

There is route to check whether some number is prime or not, by dividing it by all lower numbers.

Definition of even is that it's dividable by 2. 10 is dividable by 2, so it's even. We're not interested about any lower, or higher numbers at all.

Edited September 3, 2014 by Sensei

[Bignose]

Really? If there is a particular pattern, then let's see it proven. (And not the tautology of "the numbers in the set of primes are prime" presented in "Probability of that 2,3,5,7,11,13 etc etc. is prime number is 100%.".)

 

Using your own definition, if they aren't random, then there must be a pattern.

 

I don't know how you can say "pattern to predict the next prime is completely not needed" when that is precisely what the definition YOU CHOSE TO USE says. Not to mention that THE PATTERN ITSELF it exactly what we're talking about.

 

If this is so easy, let's see some supported results.

[Sensei]

Really? If there is a particular pattern, then let's see it proven.

Nobody is saying about pattern.

 

I just said "pattern is completely not needed".

 

If this is so easy, let's see some supported results.

bool IsPrime( unsigned long long value )

{

if( value < 2 ) return( false );

unsigned long long max = (unsigned long long) sqrt( (double) value );

for( unsigned long long i = 2; i <= max; i++ )

{

if( ( value % i ) == 0 )

{

return( false );

}

}

return( true );

}

 

If prime would be random, chance to return true or false for given input value, would be 50/50.

Instead there is returned true, if value is prime,

or false, if it's not.

 

If we would be interested about even numbers, it'd be also returning true or false,

but inside there would be:

 

return( ( value % 2 ) == 0 );

[Bignose]

Nobody is saying about pattern.

 

I just said "pattern is completely not needed".

This whole thread has been about the pattern of primes! Have you even read it? Even if you had only read the last few posts, that is still tentacle's reaction to the phrase "there is no pattern in their occurrence".

 

bool IsPrime( unsigned long long value )

{

if( value < 2 ) return( false );

unsigned long long max = (unsigned long long) sqrt( (double) value );

for( unsigned long long i = 2; i <= max; i++ )

{

if( ( value % i ) == 0 )

{

return( false );

}

}

return( true );

}

Very good, you've repeated the tautology that a number in the set of primes is prime. Because all you've done here is use the definition. The whole point about finding a pattern that predicts if a number is prime or not is to not have to use its explicit definition.

 

And the whole point in finding a pattern is to hopefully dramatically speed up the calculation time for checking a prime. Do you know how long it would take even the best computers to use the above algorithm to check the currently known largest prime number 2^57885161 - 1? Probably millenia. This is why finding a pattern is so interesting.

 

If prime would be random, chance to return true or false for given input value, would be 50/50.

Instead there is returned true, if value is prime,

or false, if it's not.

Now this I really don't get. Where does this idea that if something is random, then is has to be 50:50? This misconception has come up a lot of times, and I really don't get it.

 

Rolling a fair 6 sided die is random. None it's values come up 50:50. Dealing a deck of cards is random. The chances of being dealt a straight flush is certainly not 50:50. I could go on and on and on. There are literally an infinite number of random events out there that aren't 50:50.

 

All it takes to be random is for there to not be pattern. I mean, sheesh, look at the definition YOU POSTED!! Nothing about 50:50 in that!

 

And finally, even with the tautology and the explicit definition of primes, look at the visualizations at the beginning of this thread. No patterns. That's the whole point.

 

If you could prove a 100% accurate pattern, you'd revolutionize a great deal of mathematics. But the current state of knowledge doesn't have that. Random is exactly the right word which describes our current knowledge.

 

I really don't understand the objections to this when the objectors cannot ever produce a pattern.

Edited September 3, 2014 by Bignose

[tentacle]

You can totally disagree all you want, nonetheless, the simple truth is that with our knowledge as of right now, no pattern has been found. If you disagree, let's see proof of a pattern. No 'intuition', no 'we need a breakthrough', but an actual demonstration of a pattern. Without that, the word random is a completely valid and accurate description of our knowledge today.

 

you forgot that absence of evidence isnt evidence of absence. so i will continue to disagree. correct answer is we dont know is there pattern or not. but defenetly not : "there is no pattern" or "if you dont show pattern there is none." i hope you see difference. random isnt valid description.

 

we need to name things right names.

 

"Rose is a rose is a rose is a rose."

Edited September 3, 2014 by tentacle

[imatfaal]

 

i totaly disagree with this kind of statements. because we dont understand formula behind them that doesnt mean thy are random. beyond our chaos universe is order.

 

edit: as numero uno breaktrough in all fields of science we need math breaktrough. because with it we will have physics breaktrough.

imho, solution to prime numbers is in geometry.

 

"beyond our chaos universe is order." - citation required. In a post claiming we need to constrain definitive statements concerning unsettled areas of knowledge you sentence is a monstrous carbuncle.

 

In the context of this thread - in which I and others repeated ideas about the state of our knowledge and a ongoing search for a pattern - I will stand by what I have written. It is strange that you choose to attack based on the splitting of hairs rather than any valid concern. This thread and others are full of posts containing this sort of paragraph:

 

We do know for sure. We know that the primes are not random. It is obvious that they are a completely determined and predictable sequence of numbers.

 

And whilst I might have been more accurate (I had momentarily forgotten that this was the internet where everything had to be precise) I feel it is important to contradict and refute (if possible) the above form of statement.

 

I would also claim that "pattern" is the human or conscious recognition of an underlying regularity or predictability and as such could be claimed not to exist until identified; but that might be needlessly philosophical.

 

 

you forgot that absence of evidence isnt evidence of absence. so i will continue to disagree. correct answer is we dont know is there pattern or not. but defenetly not : "there is no pattern" or "if you dont show pattern there is none." i hope you see difference. random isnt valid description.

 

we need to name things right names.

 

"Rose is a rose is a rose is a rose."

 

If you wish to be pedantic and quibble over other posters' wording it would aid understanding if your posts were written in correct English.

 

I'll raise your Stein with a Magritte "Ceci n'est pas une pipe."

[Bignose]

you forgot that absence of evidence isnt evidence of absence. so i will continue to disagree. correct answer is we dont know is there pattern or not. but defenetly not : "there is no pattern" or "if you dont show pattern there is none." i hope you see difference. random isnt valid description.

 

we need to name things right names.

 

"Rose is a rose is a rose is a rose."

Yeah, you're probably right. Now excuse me, I'm going to go back to my work on finding a pattern in roulette table results. I know that if I just study it harder, I'll be able to find the pattern there, too. Because no one has proven that there isn't a pattern yet. I mean... 5 reds in a row, that mean the next one has almost surely got to be black!!!! Bigfoot and the Loch Ness monster have both looked over my work and think it is sound.

 

As a serious reply, maybe you should read what I have actually posted in this thread, twice now.

 

that just because we don't know a pattern today doesn't mean one doesn't exist. Nor does it mean it is useless to look for one even if one is never found.

But, again as I wrote above, until a pattern is demonstrated, your belief in one is unsupported by the best evidence and knowledge we have today. No amount of cliche-quoting, misunderstaing the word random, and wishing for a pattern to magically appear will change that. If you want to support this idea, then let's see some actual demonstration of it.

[Endy0816]

At present I'm thinking Kolmogorov complexity might apply. Basically trying to decompress some (quite possibly unknowable)constant and end up with all the prime numbers.

[PeterJ]

Regarding pattern and randomness here's how I would look at it, at the level of the basic principles. I can see an argument for various points of view.

 

The primes are an artefact of a simple mechanical process that can be described very easily. The primes, being the 'atoms' of the number line, create a combination wave of products that will determine where further primes will occur. This allows us to make basic predictions such as the 6n+/-1 rule, or predict that the products of each successive prime will reduce the density of primes higher up the line by 1/3p.

 

This process seems to be about as deterministic and predictable as a process could be. Nobody argues that the products of the primes are unpredictable. The unpredictability arises when we try to predict the location of the null points in the combination wave of the products. The individual wave-shape is identical for every prime, (if we calculate it to be so), but in combination there are so many overlapping waves, each with a unique frequency, that predicting where the gaps will occur very quickly becomes impractical.

 

In this sense the primes would be unpredictable. This is what I'd call weak unpredictability, since it is just a computational problem.

 

Because the primes are not causally linked, (p has no effect on the location of next-p), but are the output of an overarching process, it seems to me that there can never be an algorithm to predict the next one. They obviously conform to a pattern on large scales, at a statistical level, but locally they might as well be random. This would be a stronger unpredictability.

 

While I'd never expect to see a pattern in the primes besides the obvious statistical trends, I'd suggest that 'might as well be random' is not quite the same thing as 'random'. Whatever pattern they make must always be evolving and there could be no repetition of shapes for us to see, but I still feel that 'pseudorandom' (or some such word) would be a more accurate use of language than 'random'. If the product of the primes are not random, then it seems incorrect to say that the gaps in their wave-shape is random.

 

The trouble is that 'random' has various meanings, so it is difficult to say anything definite.

[Acme]

Regarding pattern and randomness here's how I would look at it, ...

The trouble is that 'random' has various meanings, so it is difficult to say anything definite.

Good gravy Peter! You seem to have no trouble saying definite various things that contradict the common understanding. Get over it.

[PeterJ]

That's not vey helpful to me. What contradictions are you referring to?

[Acme]

Something new on primes.

 

Prime Gap Grows After Decades-Long Lull

...

Now, mathematicians have made the first substantial progress in 76 years on the reverse question: How far apart can consecutive primes be? The average spacing between primes approaches infinity as you travel up the number line, but in any finite list of numbers, the biggest prime gap could be much larger than the average. No one has been able to establish how large these gaps can be.

...

This past August, two different groups of mathematicians released papers proving a long-standing conjecture by the mathematician Paul Erdős about how large prime gaps can get. The two teams have joined forces to strengthen their result on the spacing of primes still further, and expect to release a new paper later this month.

...

Erdős conjecture is based on a bizarre-looking bound for large prime gaps devised in 1938 by the Scottish mathematician Robert Alexander Rankin. For big enough numbers X, Rankin showed, the largest prime gap below X is at least [math]{\Large \frac{1}{3} \frac{\log X \log\log X\log\log\log\log X}{(\log\log\log X)^2}}[/math]

...

Erdős made a more modest conjecture: It should be possible, he said, to replace the 1/3 in Rankins formula by as large a number as you like, provided you go out far enough along the number line. That would mean that prime gaps can get much larger than in Rankins formula, though still smaller than in Cramérs.

 

The two new proofs of Erdős conjecture are both based on a simple way to construct large prime gaps. A large prime gap is the same thing as a long list of non-prime, or composite, numbers between two prime numbers. Heres one easy way to construct a list of, say, 100 composite numbers in a row: Start with the numbers 2, 3, 4, , 101, and add to each of these the number 101 factorial (the product of the first 101 numbers, written 101!). The list then becomes 101! + 2, 101! + 3, 101! + 4, , 101! + 101. Since 101! is divisible by all the numbers from 2 to 101, each of the numbers in the new list is composite: 101! + 2 is divisible by 2, 101! + 3 is divisible by 3, and so on. All the proofs about large prime gaps use only slight variations on this high school construction, said James Maynard of Oxford, who wrote the second of the two papers.

...

[Unity+]

Yeah, you're probably right. Now excuse me, I'm going to go back to my work on finding a pattern in roulette table results. I know that if I just study it harder, I'll be able to find the pattern there, too. Because no one has proven that there isn't a pattern yet. I mean... 5 reds in a row, that mean the next one has almost surely got to be black!!!! Bigfoot and the Loch Ness monster have both looked over my work and think it is sound.

An example of a problem that is assumed by many scientists to be false is the P vs. NP problem. Although it is unproven, scientists continue as if it is false. Whether this is the best way to go about things is up for another discussion. However, I think the realization would be that concepts with seemingly random constructs have some pattern behind them, one way or another.

[Bignose]

An example of a problem that is assumed by many scientists to be false is the P vs. NP problem. Although it is unproven, scientists continue as if it is false. Whether this is the best way to go about things is up for another discussion. However, I think the realization would be that concepts with seemingly random constructs have some pattern behind them, one way or another.

Look, I don't care what assumptions, realizations, conceptualizations, or constructions you want to make. The simple truth is that no definitive pattern has been found. That is the state off our best knowledge today. To say otherwise is a falsehood. Once again, if you disagree, just show us the pattern.

 

All this other stuff is immaterial to this point. All your personal opinions on how likely that there is a pattern is immaterial. All my personal opinions are immaterial. The current state of our knowledge is known. Why do you continue to argue it?

[Unity+]

Look, I don't care what assumptions, realizations, conceptualizations, or constructions you want to make. The simple truth is that no definitive pattern has been found. That is the state off our best knowledge today. To say otherwise is a falsehood. Once again, if you disagree, just show us the pattern.

 

All this other stuff is immaterial to this point. All your personal opinions on how likely that there is a pattern is immaterial. All my personal opinions are immaterial. The current state of our knowledge is known. Why do you continue to argue it?

I don't know why you are getting defensive, I was simply stating a point.

 

EDIT: Couldn't we bring the Sieve of Erathosthenes into question in regards to its state of randomness?(Please don't attack me for this, it's just a question).

Edited December 27, 2014 by Unity+

[PeterJ]

I cannot see how there could ever be an overall pattern given the mechanism that creates the primes. But there is some pattern. The density falls as the numbers grow, and the primes always occur next to 6n. This is surely a pattern. But that's about it. The combination wave of the prime products is eternally evolving so the details of the pattern are never settled. All the same, it seems wrong to say there is no pattern at all.

[Unity+]

I cannot see how there could ever be an overall pattern given the mechanism that creates the primes. But there is some pattern. The density falls as the numbers grow, and the primes always occur next to 6n. This is surely a pattern. But that's about it. The combination wave of the prime products is eternally evolving so the details of the pattern are never settled. All the same, it seems wrong to say there is no pattern at all.

The only problem(and this is after thinking about the problem after a while) with this assumption is this line of thinking could apply to any line of thinking in regards to any considerably random line of elements, i.e. "Finding something that isn't there"

[PeterJ]

Hi Unity. What assumption was that? I didn't think I made any.

[Unity+]

Hi Unity. What assumption was that? I didn't think I made any.

 

 

I cannot see how there could ever be an overall pattern given the mechanism that creates the primes. But there is some pattern.

The assumption is that there is a pattern when there is no overall pattern.

[PeterJ]

Oh okay. I thought the comment was aimed at me. It's not an assumption but a fact that there are detectable patterns. I did say that there can never be an overall (comprehensive) pattern, but there are certainly patterns that can be detected. The prime number theorem describes one of them.

[Unity+]

Oh okay. I thought the comment was aimed at me. It's not an assumption but a fact that there are detectable patterns. I did say that there can never be an overall (comprehensive) pattern, but there are certainly patterns that can be detected. The prime number theorem describes one of them.

Well, then Bignose and you may be on two different pages then. I think he is referring to overall patterns. The same thing could be said about Pi, where there may be patterns in one part of pi, but overall there is none.

[PeterJ]

Hmm. That's not a similar case. By 'pattern' I would mean a pattern that is predictable and explicable.

[Unity+]

Hmm. That's not a similar case. By 'pattern' I would mean a pattern that is predictable and explicable.

Well then we aren't on the same page either, it seems.