Do Infinitely Many Prime Pairs Exist?

[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.

 

[Spin_Network]

My work in prime numbers has delivered me evidence that prime numbers are not connected to each other due to their prime nature, but due to the connections of the other numbers in prime number locations (which happen to be multiplications of prime numbers). I wrote extensively about this in Chapter 5 of In Search of a Cyclops http://www.pentapublishing.com/CyclopsBook5.html . Published as The Proof of Nothing in the year 2000.

When dividing all numbers in series of six it is possible to see that there are prime number positions and non-prime number positions. All prime numbers (except 2 and 3) appear in either the first position or in the fifth position. See http://www.pentapublishing.com/images/table1.gif
I made the choice to place 1 with the prime numbers even though that is not considered correct. The link between prime numbers is not a link between the prime numbers, but a link between the non-prime numbers. See http://www.pentapublishing.com/images/table2.gif
What are the red numbers in this table are non-prime numbers in first and fifth positions. These red numbers follow a pattern, like for instance those divisible by 7. See http://www.pentapublishing.com/images/table4.gif
As you can see in this table 7 'cuts' out numbers in turns in first and fifth positions according to a pattern of 4 plus 3 lines of six down.
All prime numbers 5 and up 'cut' away first and fifth position numbers from the prime number list according to a very strict and specific regimen. See http://www.pentapublishing.com/images/jump.gif which shows two numbers after each prime number (or multiplication of a prime number), portraying the lines of six jumped to take out numbers off the prime number list. All prime number multiplications follow these kinds of number specific jumps - and all these jumps are linked together. Therefore all prime numbers can then be calculated according to a subtraction method. It is not the link between them, but the link that eliminates the others from being prime numbers. From this information a new method can be created to quickly appoint the prime numbers for which I am in the (long) process of getting a patent (question so far is if that will happen).

Once you understand how non-prime numbers are formed, it becomes obvious (and a little bit boring) to see which numbers remain as the prime numbers. All prime numbers would appear in two packs if it wasn't for the multiplications of prime numbers taking numbers off that list of prime numbers. Have a look at http://www.pentapublishing.com/CyclopsBook5.html , a Chapter I wrote to deliver evidence that zero is intrinsic to the natural numbers (not a natural number itself according to number theorists - but which is a natural number according to set theorists).

Whatever you are trying to convey, in some sort of amazing discovery?

Take this:http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm

I discovered that the 'Ero-sieve' is not anatural representation of the numbers used, if one diagonalize's the cells into a certain progressive angle, then certain patterns arise.

I did not know at the time that this pattern is actually pertaining to:http://eureka.ya.com/angelgalicia30/Primesbehaviour.htm

which is quite an amazing site!

One can see that Prime numbers behave in a specific way?..if one uses the 'rectangle Sieve' for the first one hundered primes, there is NO concerning pattern that emerges. But if one Diagonalize's the sieve boxes, then one see's the Pattern is Emergent!

What is the significance?..it happens to be connected to Fractals and Prime Number distribution of real numbers, and my lack of Mathematical skills gave me a false sense of achievement ..the process was allready in existence, with maybe a slight of hand!

 

[shmoe]

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.

Same thing. In a basic Eratosthenes of size N you'll store your information in a string of N bits, where the nth bit is 1 if n is prime, 0 if it's composite. You start with a string of all 1's, 111111... First stage you remove 1: 011111... Next stage multiples of 2:0110101010... then multiples of 3:01101010001.. And so on.

Presieving by 2 and 3 is not only to save time but space. We need only consider numbers congruent to 1 or 5 mod 6, so this is the sequence we store. We need only about N/3 bits, the first bit represents 1, the next 5, the next 7, the next 11, then 13, and so on. Each pair of bits is one of your rows, so I'm going to separate them with commas. First stage we knock off 1: 01,11,11,11,11,11,... Next stage we knock off multiples of 5, this isn't as straightforward due to how our string is indexed, but we can easily see that we need to jump ahead 4 pairs and remove the first in the pair (this is 25) then we jump ahead one pair and remove the second (35) to get: 01,11,11,11,01,10,...And we continue jumping 4 pairs, 1 pair, 4 pairs, 1 pair crossing off as we go. Replace pair with line and this is what you're describing. I described in an earlier post how you'll know how many pairs (used your line terminology) to jump based on the k and the 1 or 5 in your primes representation as p=6k+1 or p=6k+5. Here k is just indexing the pair (or line) you're in (starting at 0) and 1 or 5 is whether you are 1st or 2nd in the pair.


A nice thing about Euclid's proof is that it's somewhat constructive and you can phrase in such a way to avoid the troublesome "infinite". If you take *any* finite list of primes then it not only guarantees there is a prime not on your list but also gives a multiple of this new prime, so you could find it with a little factoring. If you start with 3 and 5 it gives N=3*5+1=16, which though not prime has a the prime divisor 2, not on our list. We now take 2,3,5 and make N=2*3*5+1=31, another prime not on our list. So, when you've gotten to the point where you think you have all the primes, find the corresponding N and then what? What in your "reality" goes wrong? Did you just find a number with no prime factors? (do you think that's possible?)

 

[Fredrick]

Please, stop posting such nonsense, for the love of mathematics.

Thank you, guys, for very good answers, and good information.
I do not try to undermine math, math is fine as it is. I was just answering a question, and bumped into the limitations. You may think the limitations are inside of me (and you may be right), I think the limitation exist in math (in that it is abstract only, despite its many realistic applications). The thread divulged into a question about the infinite, which does not belong in this thread.
My famous last words:
I think the infinite in math is a mirror's mirror's image that is being given value. The infinite image truly exists — we need not argue about that — but is it still reflective of reality?
Thanks again.

 

[Gokul43201]

My famous last words:
I think the infinite in math is a mirror's mirror's image that is being given value. The infinite image truly exists — we need not argue about that — but is it still reflective of reality?

With that, let's call it a day here, wot ?