Do Infinitely Many Prime Pairs Exist?

[shmoe]

Not sure what that is... let me look it up... Oh, I see. So no big deal that they can be expressed in prime numbers;

Well the fundamental theorem of arithmetic is a huge deal, but yes, it's nothing specifically special about these gaps.

but most of the time they are a single prime (as you stated above) and the number 2 is never used, so there is a slight difference.

I don't believe I said that, only that all (possible) gaps are conjectured to occur infinitely often. The actual conjectured frequences are related to the odd prime factors of n+1 (in your notation).

 

[Rahmuss]

... If all these gaps were somehow randomly chosen odd numbers from 1 to 17, you'd expect about 6/9 of them to be prime, which turns out to be close to the truth...

That's what I meant as far as you saying how many of them would be prime; but I see now that you were specifically stating that case for the numbers from 1 - 17.

And I agree that the fundamental theorem of arithmetic is a big deal. I just meant as you stated later, that the gap between them being primes is not as big a deal as I thought it might be.

Has anyone really tested these ideas in a different based number system? Either switching the current 10-based primes to a different based number system, or finding the prime numbers in a different based number system?

 

[Rahmuss]

Is this number a prime number?

35892379... many many more numbers ... 234127

 

[Fredrick]

Reasonable answers, but if you agree then what is this whole thread about?

I don't understand what you think is new or worthy of a patent.

The sieve of Eratosthenes is rather slow. My method based on a link that exists in between the sieve of Eratosthenes is rapidly quick, even with the larger numbers (though my computer only allows me to go to one hundred billion (yes, all primes captured), and my requests to use super computers were all for naught.) While getting all prime numbers takes up time, getting the primes between 99,999,999,000 and 100,000,000,000 is perky quick. Prime numbers are the basis for securing online banking; I don't know, maybe it can be the motor for some impossible to crack codes.

That your "jump.gif" chart has a pattern that can apparently go backwards nicely is only because your table is too short.

The 'etcetera" at the bottom is not misleading in that the pattern is not based on prime numbers, but on figuring out which numbers are multiplications in first and fifth positions. I am not interested in why the prime numbers are what they are, but why the other numbers in first and fifth position are not prime numbers. So, the patterns in the "jump.gif" continues indefinitively (prime number or not), while knocking numbers off the prime number list. Number 25 is only a redundacy on this list (because it is a multiplication of 5 and knocks off the exact same numbers as 5 already took off the list). Yet number 25 is part of the patterns, and helps establish correct patterns that follow.

Including 0 as a natural number or not is totally irrelevant to how any number theorist will think.

I agree. I just had to mention it, because the links I used belong to my book that establishes reasonable (mathematical) evidence that a unified field of forces cannot exist, since a platform on which a theory of everything can be placed has to include the force of separation as well. While nothing is just plain nothing, there is a function attached to it; very much like the zero in the binary system is crucial.
A definition is simply just that, a definition, but the existence of zero in a certain spot in one definition (with set theory) and in a different spot in another definition (in number theory) gives insight into the trivial nature of definitions. As such it establishes that it is part of the human aspect of mathematics (where most people do not expect any human interference).

The same goes for putting 1 in it's own special category. The fact that it has a multiplicative inverse in the integers makes it fundamentally different from everything else.

Correct. It is a trivial definition for which some evidence can be given for, and some evidence can given against. As you can read in my chapter 5 I have no problems with definitions and various outcomes. I rebut 1 as a prime later in this chapter. 1 is a very special number.

My counter question:
Why this thread (of five pages long already) if all you are curious about is the fact that prime numbers often (especially in the beginning) appear in couples. It is so simple.

Imagine a bowling lane. The prime numbers are the pins that remained standing after a ball ran through the other pins. Prime numbers are numbers that had nothing happening to them. And why are they paired in two's? Because a bowling center has multiple lanes. The chance that the first and the last pins remain standing on a lane is - on average - better than any other pattern to form.
In lane One the last pin remains standing, while in lane Two the first and the last pins remain standing, where in lane Three the first but not the last pin remains standing. Voila, a pattern has been established where we see a pairing up of primes with a last pin standing on one lane close to a first pin standing on the next lane. It has nothing to do with the pins themselves, but everything with their location and the existence of separate lanes. The lanes in this particular indefinite bowling center have 6 pins each, while the number of balls thrown per lane slowly increases when moving up along the lanes; taking more pins out the further away the lane is from the start.

 

[shmoe]

The sieve of Eratosthenes is rather slow. My method based on a link that exists in between the sieve of Eratosthenes is rapidly quick, even with the larger numbers (though my computer only allows me to go to one hundred billion (yes, all primes captured), and my requests to use super computers were all for naught.) While getting all prime numbers takes up time, getting the primes between 99,999,999,000 and 100,000,000,000 is perky quick. Prime numbers are the basis for securing online banking; I don't know, maybe it can be the motor for some impossible to crack codes.

Eratoshtenes is slow?? Have you looked around for various implementations? I have a pretty basic one on my computer that can spit out all primes up to 10^9 in 52 seconds (athlon 3000+), up to 10^8 in 4 seconds. You might check out http://wwwhomes.uni-bielefeld.de/achim/prime_sieve.html who claims to be able to produce all primes up to 10^9 in under 52 seconds on a crappy 133Mhz pentium. Neither of these is even using the most sophisticated sieving techniques available.

What counts as "perky quick"? The built in primality testing function of maple V can find all primes in the range you've given before you can think "boo". Have you actually looked into the current technology and compared your algorithms speed?

The 'etcetera" at the bottom is not misleading in that the pattern is not based on prime numbers, but on figuring out which numbers are multiplications in first and fifth positions. I am not interested in why the prime numbers are what they are, but why the other numbers in first and fifth position are not prime numbers. So, the patterns in the "jump.gif" continues indefinitively (prime number or not), while knocking numbers off the prime number list. Number 25 is only a redundacy on this list (because it is a multiplication of 5 and knocks off the exact same numbers as 5 already took off the list). Yet number 25 is part of the patterns, and helps establish correct patterns that follow.

So you are removing multiples of *all* numbers congruent to 1 or 5 mod 6? In this case the pattern will continue, but it's a trivial observation like I've mentioned and one that is exploited in a basic sieve that "pre-sieves" by 2 and 3, that is to say they only consider numbers congruent to 1 and 5 mod 6. This is the first "obvious" improvement on the basic Erathosthenes. However they usually wouldn't include redundancies like crossing off multiples of 25 since they are already removed, so I don't see how you have anything that resembles an improvement?

My counter question:
Why this thread (of five pages long already) if all you are curious about is the fact that prime numbers often (especially in the beginning) appear in couples. It is so simple.

Cripes, if you think it's so simple how about you produce a proof there are infinitely many prime pairs? I guarantee instant fame, at least amongst mathematicians. Especially if you talk about bowling lanes rather than congruence classes.

 

[Fredrick]

It is simple.

Cripes, if you think it's so simple how about you produce a proof there are infinitely many prime pairs? I guarantee instant fame, at least amongst mathematicians. Especially if you talk about bowling lanes rather than congruence classes.

I have two answers. The first one is just words, the second contains a view. And, no, you will not get more math out of me than I have already given. This is as specific as I get.

1/ Eratosthenes already gave the answer what the primes are and how they come about. I just found a quicker way in delivering what he already delivered. The original answer has been given more than 2200 years ago. There are no important questions left about the primes.

2/ To find out if there are infinitely many prime pairs one need to understand that the bowling lanes are side by side infinitely. While looking at what is happening to the first couple of billion bowling lanes it becomes obvious that the further away from the start the less often the pairing takes place because more and more balls are thrown per lane of six pins, therefore improving the chance all six pins are kicked down. The kicking down is an increasing process, but from some respect it is only increasing in intensity, not always in effectiveness.

As you already pointed out the pattern of 25 kicking down pins further down the line is as helpful as cleaning a clean glass. The further down the road we look, the more balls are generated following the exact same track as previous balls, so more balls does not equate to more effectiveness. Only primes will kick down pins further down the line according to a new pattern, and the occurrence of primes diminishes. The reason their occurrence diminishes is that the basis for pins to be kicked off the bowling lanes is based on the pins that remained standing on previous lanes. There is no reasoning around this, so let me repeat it here: the reason pins are kicked down the bowling lanes is based on the pins that remained standing on previous lanes. Pin number 5 remained standing, which resulted in pin number 25 to be kicked down. The chance for a pin to remain standing gets diminished by the previous occurrence of a pin in the same location. This chance never diminishes, it only increases.

No matter how long it will take, somewhere in the infinite, we will run out of primes altogether, because each newly discovered prime will start taking down its own pins according to its own pattern further down the line. So even the single prime that was able to escape each and every other pattern will be the very reason for establishing a pattern that ensures it won't happen again. It will be way way out somewhere in the infinite, but primes cease at one point to exist — inherently. The reasons it happens way way out in the infinite can be found at the beginning: it took the first pin that remained standing (5) all the way till 25 to take out a similar pin according to the pattern of that first pin. Not until 25 was a pin kicked down in that location. The primes therefore got an enormous head start, and even with new quicker methods it may remain too far out for us to finally realize the last prime was found (I don't expect it before the year 2140). Yet the kicking down is inherent. Inherent meaning, if primes don't cease to exist they themselves become the reason following primes in the same pattern cease to exist.

All six pins will be kicked down all the time from one point on. But could the last prime found be part of a prime pair? You know what? I think it is.

So you see, no instant fame for me, because proof that there are infinitely many prime pairs is proof that cannot be found, because it does not exist.

P.S. The patent I am trying to acquire is not based on the sieve of Eratosthenes but on appointing prime locations in consecutive lines of six. No calculations are done on numbers, only on (positions in) lines of six.

 

[shmoe]

Please, enough with the bowling lanes already. There already exists simple and clear language for discussing this (congruence classes and such), how about you try to use that?

"All six pins will be kicked down all the time from one point on."

No, there are infinitely many primes, we never run out and so there will always be "pins" left standing no matter how far out you look. Euclid's proof is pretty standard, you might want to look into it.

If the thing you're trying to patent is anything like you're attempting to describe with bowling lanes, it is most definitely based on Eratosthenes. Have you ever looked at the sieve after "pre-sieving" by 2 and 3? This is exactly what you're describing (but with redundancies).

"There are no important questions left about the primes."

Haha. You're joking right? Do you really believe Eratosthenes sieve answers everything? The prime number theorem wasn't important? Improving its error term isn't important?

 

[shmoe]

Has anyone really tested these ideas in a different based number system? Either switching the current 10-based primes to a different based number system, or finding the prime numbers in a different based number system?

The base is pretty much irrelevant, the base is just how you represent numbers, it has no bearing on whether they are prime or not.

Is this number a prime number?

35892379... many many more numbers ... 234127

You mean filling in with some digits? 358923790000000000000000000234127, 35892379123154452234127? If so, the answer is sometimes yes, sometimes no (in that order for these examples).

 

[Hurkyl]

P.S. The patent I am trying to acquire is not based on the sieve of Eratosthenes but on appointing prime locations in consecutive lines of six. No calculations are done on numbers, only on (positions in) lines of six.

It's still nothing new -- people already sieve on far more complicated things.

Specifically, you're talking about line sieving on the two functions:

6n - 1
6n + 1

In the general case, you sieve on an arbitrary polynomial f(n). You know that if p | f(n), then p | f(n + p), so this let's you do an ordinary sieve on some interval [a, b] to find out what numbers in the image f(a), ..., f(b) factor.

One important example is the quadratic sieve, used by the quadratic sieve factoring algorithm. It sieves on specific quadratic polynomials looking for numbers in the range of the polynomial that have only "small" prime factors.

it may remain too far out for us to finally realize the last prime was found

Are you seriously claiming there are only finitely many primes?

 

[Hurkyl]

Incidentally, you might want to look into the "Wheel sieve".

 

[Hurkyl]

Actually, I'm wrong -- you haven't quite gotten to the effectiveness of line sieving on 6n+1 and 6n-1: it appears you still want to allocate space for all numbers, but only work along those two lines.

It is more efficient to allocate only the space you intend to use. E.G., for sieving over 6n+1, you would:

Allocate an array T = [a, a+1, ..., b]
For each prime p:
Find the first number k in [a, b] such that 6k + 1 is divisible by p.
Cross off k, k + p, k + 2p, ... in T.
Go through the entries of T, and for each x that isn't crossed off, print 6x + 1.

Your "jump pattern" reduces to using the k you found when sieving 6n+1 to figure out the k you need when sieving 6n-1. (And yes, this idea is also already known -- e.g. look at optimizations of the multiple polynomial quadratic sieve)

 

[mathwonk]

why do these [questionable] threads get the most attention? easier access? like number theory itself?

 

[Hurkyl]

"Defenders of the orthodoxy" vs "lone point of light" and all.

Often times, I learn from them -- for instance, I learned about wheel sieving from this thread, so it was worth it for me, in that respect.

 

[Fredrick]

You placed me well!

Euclid's proof is pretty standard, you might want to look into it.

There is multiple proof out there about primes into the infinite. I know that. I consider them mirror images of mirrors, the calculations are correct, but are the grounds for the calculations correct? I think the infinite calculations are horizon calculations, and would even show that the Earth is flat ad infinitum. I may be kidding about the year 2140, but I think in reality primes are limited, and the evidence is not going to become clear with most people going for Euclid's proof (etc) until we actually run out of them; which has not happened yet.

If the thing you're trying to patent is anything like you're attempting to describe with bowling lanes, it is most definitely based on Eratosthenes. Have you ever looked at the sieve after "pre-sieving" by 2 and 3? This is exactly what you're describing (but with redundancies).

I'll look into that, thank you.

"There are no important questions left about the primes."

Haha. You're joking right? Do you really believe Eratosthenes sieve answers everything? The prime number theorem wasn't important? Improving its error term isn't important?

Once you see what primes are, what else is there, except for lesser stuff (even when the lesser stuff is more complicated)?

 

[Hurkyl]

but are the grounds for the calculations correct?

The hypotheses for the theorem are nothing more than the definition of the natural numbers and its arithmetic. So yes, the grounds for the theorem are correct.

 

[Fredrick]

The hypotheses for the theorem are nothing more than the definition of the natural numbers and its arithmetic. So yes, the grounds for the theorem are correct.

But will they deliver reality or do they deliver a clever mathematical out that is not real?

 

[shmoe]

There is multiple proof out there about primes into the infinite. I know that. I consider them mirror images of mirrors, the calculations are correct, but are the grounds for the calculations correct? I think the infinite calculations are horizon calculations, and would even show that the Earth is flat ad infinitum. I may be kidding about the year 2140, but I think in reality primes are limited, and the evidence is not going to become clear with most people going for Euclid's proof (etc) until we actually run out of them; which has not happened yet.

Have you actually read Euclid's proof? It takes any finite number of primes, p_1,\ p_2,\ldots,p_k, and produces a new number N=p_1p_2\ldots p_k+1 which is not divisible by any of them. But we know that N must have a prime divisor, so there must be a prime that you left out. Now come 2140 if you think you've got all the primes, just multiply them together, add 1, and factor-voila, another prime to be had (note N itself may be the new prime).

Once you see what primes are, what else is there, except for lesser stuff?

Umm, have you looked at any the vast body of research that exists in prime number theory? I think you're alone in thinking that the questions that aren't answered by the basic sieve can be classified as "lesser stuff". The prime number theorem is a huge fat one that comes to mind. Pick up any decent number theory text and do some investigating on what's happened since eratosthenes.

 

[Hurkyl]

But will they deliver reality or do they deliver a clever mathematical out that is not real?

Reality, of course. The theorem applies to anything satisfying the definition of the natural numbers and its arithmetic.

An example of something that satisfies the definition of the natural numbers and its arithmetic is the natural numbers, and its arithmetic.

So, the theorem applies to the natural numbers, and its arithmetic.

 

[shmoe]

why do these [questionable] threads get the most attention? easier access? like number theory itself?

I think it really is easy access. Many problems in number theory can be so easy to state that you can explain them to children. Laymen can then fantasize that these simple looking problems also have simple solutions that generations of brilliant mathematicians have overlooked. This is a blessing as it's easy to explain the sorts of things number theorists work on, a curse because of all the clutter produced by "proofs" of fermat's last theorem, goldbachs, etc.

Another common occurrence is people just not reading even the basics of what's known and then believing that they've stumbled upon something new and wondrous. While it's great that people like to delve into some number theory for fun, it can be hard to get them to realize that what they've done, while possibly wondrous, is certainly not new. It's even harder to convince someone their little pet theory is new but in fact wrong. (I'm not directing this at anyone in particular in this thread, just my general experience)

 

[Fredrick]

Reality, of course. The theorem applies to anything satisfying the definition of the natural numbers and its arithmetic.
QUOTE]
Math is not reality, its an abstract. Question is in how far is math representative of reality in the realm of the infinite!

 

[Hurkyl]

Math is not reality, its an abstract.

And the reality of the matter is that mathematical things (like the natural numbers) play by mathematical rules.

If you'd like to argue that, aside from the abstraction, the natural numbers have no place in reality, you're free to do so in the appropriate place. (The philosophy of science & mathematics forum, for example)

Question is in how far is math representative of reality in the realm of the infinite!

Mathematical things play by mathematical rules, as I said above. Of course, mathematics will say nothing about ill-formed ideas that constantly morph to avoid counterarguments! (Which is what many laypeople mean by "infinity")

 

[Fredrick]

And the reality of the matter is that mathematical things (like the natural numbers) play by mathematical rules.

That supports my point, thank you.
I have no problem with math being played out in the abstract at all. It is the beauty of math. A single mirror still reflects reality. Yet when an abstract (like the infinite) is placed within the abstract of math, a mirror within a mirror is created delivering a view no longer solely based on reality but on its reflective capabilities as well.
When looking at the number of primes between 1 and 100, there are 25 primes. When looking at the first hundred numbers after ten thousand (10,000 because the active working of primes to take out first and fifth positions as prime only starts after their square), there are 11. When looking at the first hundred after one hundred million, there are 6, When looking at each subsequent hundred in following similar multiplied locations, the number of primes diminishes - big time! The option for prime numbers to appear gets eaten up by each newly found prime number. The options are limited (though truly enormous). Not today, but we are running out of them.
I have great respect for Euclid, but I think he and others got lost inside the math mirror on this one. While normally math may test reality,and delivers much information and insight, I believe that prime numbers will run out, and this will be a case where reality will test math, and deliver information and insight to math.

 

[Hurkyl]

You really don't understand the point of anything I said. This is the whole point of the axiomatic method:

If you accept the axioms, you must also accept the consequences of those axioms.

This isn't just in mathematics -- it applies to any school of thought that employs logic. If you accept the hypotheses of a valid argument, you must accept the conclusion.


Of course, this is all assuming you wish to remain logically consistent -- it's somewhat more difficult to reason with someone who insists on being irrational.


And guess what? You can't hide behind your misconceptions about the infinite for this one -- Euclid's proof demonstrates that the hypothesis "There exists a largest prime" leads to a logical contradiction... exactly the claim you are making.

 

[Hurkyl]

When looking at the number of primes between 1 and 100, there are 25 primes. When looking at the first hundred numbers after ten thousand (10,000 because the active working of primes to take out first and fifth positions as prime only starts after their square), there are 11. When looking at the first hundred after one hundred million, there are 6, When looking at each subsequent hundred in following similar multiplied locations, the number of primes diminishes - big time! The option for prime numbers to appear gets eaten up by each newly found prime number. The options are limited (though truly enormous). Not today, but we are running out of them.

Have you read the prime number theorem? It directly explains this observation of yours (yet does not predict that the primes "run out").

And then there's the theorem that, for any positive n, there exists at least one prime number p in the range n \leq p \leq 2n.

 

[Fredrick]

You really don't understand the point of anything I said. This is the whole point of the axiomatic method: If you accept the axioms, you must also accept the consequences of those axioms.

I am sorry, I am misunderstanding you right now. While I can see the abstract of the prime numbers because I can really see them in reality (5 appels can only be divided in whole appels by giving them to 5 people or to 1 person when the end results needs to be a balanced end result), but the infinite is just an abstract too far. Its construction is possible in math and as such I fully state that the calculations are correct. A Picasso painting is also really a painting. But the grounds - not the grounds of math but the grounds of reality - are not followed. So in reality I can find the primes, but in reality I cannot find the infinite where the primes are concerned.

 

[Moo Of Doom]

Reality does not exclude the possibility that there are infinite primes, reality only deems that we can never know ALL of them. In that sense, yes, we will only see a finite number of primes, but you must accept that there is always another one, and another one, and another vigintillion greater than them.

 

[Fredrick]

Reality does not exclude the possibility that there are infinite primes, reality only deems that we can never know ALL of them. In that sense, yes, we will only see a finite number of primes, but you must accept that there is always another one, and another one, and another vigintillion greater than them.

In math I accept, in reality I do not.

 

[Fredrick]

Have you ever looked at the sieve after "pre-sieving" by 2 and 3? This is exactly what you're describing (but with redundancies).

I think it is something different. But please let me know if you have seen this all before. Prime number 5 creates a pattern of 1 + 4. Starting out at the six line that begins with 25 ONE line is moved down, while going from first to fifth location. Here, 35 is found. FOUR lines down moving back to first place, 55 is found. ONE line down moving to fifth location, 65 is found.

For prime number 7 there is a pattern of 4 + 3. At the square, take FOUR lines down, swap to fifth position, and you'll find 77, Move THREE lines down, swap to first place, you'll find 91.

Eleven has a pattern of 3 + 8, and I already know what pattern 13 has because the last number of 11 (8) is repeated in first place for the pattern of 13 (8 + 5). The total is also 13, so I now know also 17's pattern (5 + 12), etcetera. The pattern is based on divying up the natural numbers in lines of six.

 

[Hurkyl]

While I can see the abstract of the prime numbers because I can really see them in reality (5 appels can only be divided in whole appels by giving them to 5 people or to 1 person when the end results needs to be a balanced end result), but the infinite is just an abstract too far.

An inability to comprehend something does not make it false.

 

[matt grime]

"creates a pattern"?

what pattern does it create according to what rules, and what does the existence of this pattern imply? That there are a finite number of primes? Please, stop posting such nonsense, for the love of mathematics.