Do Infinitely Many Prime Pairs Exist?

[shmoe]

And all of the prime numbers becomes the MAIN divisor for the first time when 6x+1 is the square of that prime number..

What exactly is a "main divisor"?

I also have no idea what you are talking about in the rest of your posts. 5 is a divisor of 6(1)-1, so why isn't 5 "added to the list" at this point? Could you perhaps elaborate on what you are trying to say?

 

[ExecNight]

Shmoe you should read my other prime post

To see why 7*5 is not added

MAIN divisor means its lowest divisor for example


77 : 11 is divisor, 7 is divisor but 7 is the MAIN divisor because it is the lowest possible number that can divide 77...

or

287 : 41 is divisor but as you can see above 7 is the MAIN divisor again..

 

[shmoe]

MAIN divisor means its lowest divisor for example

You mean lowest divisor besides 1? You know this isn't a standard term. If you want to use your own terminology and expect people to understand you it's a good idea to provide your definitions.

So it seems that 7 is the main divisor of 7, why wouldn't you add 7 to your list at 7 then? (I had meant 7, not 5 in my last post, whoops) Or does "main divisor" mean "smallest divisor of a number other than itself or one"?

Shmoe you should read my other prime post

Do you mean this one https://www.physicsforums.com/showthread.php?t=38437 ?

 

[ExecNight]

Shmoe i understand why you don't understand..And that is because you are not thinking in the ways of an algorithm.

As you can see your observations has no computational use..And you still show me states which my algorithms first few lines has already eliminated. if you still don't know what i am talking about here it is again then;

y=1
y increases y+1=2 and so on..

IF x = 5y-1 then
6x+1 is not a prime

IF x = 5y+1 then
6x-1 is not a prime

As you can see above, The number 35 was eliminated by the algorithm already..So it primality is out of question already,

Number "1" divides all, so it is absurd to take it into consideration..

And why i don't take 7 at 7 is because it is a prime number, i am talking about non-primes that breaks the 6x+1, 6x-1 rule. Before making comments you should first look what I am talking about...

 

[shmoe]

Shmoe i understand why you don't understand..And that is because you are not thinking in the ways of an algorithm.

As you can see your observations has no computational use..And you still show me states which my algorithms first few lines has already eliminated. if you still don't know what i am talking about here it is again then;

How about you actually write out your algorithm clearly then if you expect people to understand it.

It looks like you are hoping to cross of composite numbers n by checking for prime divisors that are less than sqrt(n) (this is a generous interpretation). If this is the case, why on Earth don't you just say this is the sieve of Erathosthenes (after "pre-sieving" 2 and 3)rather than make up your own terms and presenting things in such an incomplete and obscure way as you have here.

 

[ExecNight]

10 y=1, x=1
20 If 5y-1<x Go to 60
30 If 5y+1<x Go to 70
40 If 5y-1=x Go to 90
50 If 5y+1=x Go to 100
60 Print "6x+1" is a possible prime
70 Print "6x-1" is a possible prime
80 x=x+1 Go to 20
90 Print "6x+1" is not a prime it is divisible by 5. Go to 50
100 Print "6x-1" is not a prime it is divisible by 5
110 y=y+1 Go to 80


I am waiting for your comments on the algorithm Shmoe, Thanks...

 

[shmoe]

It has several problems, but from the looks of it you are just trying to remove multiples of 5 from the sequences 6n+/-1? You've included nothing about adding numbers to the list of divisors, which is where most of the confusion was.

If all you want to do is knock of multiples of 5 from the sequences 6n+/-1 there is a much simpler way to do it. Just use a divisibility test. If 6n+/-1 is divisible by 5, chuck it, otherwise move on.

Alternatively, if you've already generated the sequences 6n+/-1, and they're in order, say like 5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55..., you can remove 25, go 3 down the list, remove 35, go 7 down the list, remove 55, go 3 down the list, etc. Keep alternating between removing the 3rd and the 7th on the list and this will strike off every multiple of 5 (this pattern follows from where multiples of 5 that aren't divisible by 2 or 3 live mod 30).

Or you could instead consider all numbers of the form 30n+/-1, 30n+/-7, 30n+/-11, and 30n+/-13. All primes greater than 5 will be of this form (of course not every number of this form is prime) and none of the numbers of this list will be divisible by 5 (or 3 or 2). (this isn't really much different from the last paragraph though)

 

[ExecNight]

well shmoe if i had found a rule when the prime divisors join, i would have alrady created a perfect formula that always generates a prime number.

 

[shmoe]

well shmoe if i had found a rule when the prime divisors join, i would have alrady created a perfect formula that always generates a prime number.

I guess I'm totally confused as to what you are trying to accomplish. As it stands your algorithm appears to be an attempt to find the numbers of the form 6n+/-1 that are not divisible by 5 (in other words numbers not divisible by 2, 3, or 5). Is this not a correct assessment?

Or are you considering this some kind of incomplete algorithm for finding all primes (up to a certain finite point I'd assume)? If so have you investigated and understood the simple sieve of Erathosthenes? Do you understand how it determines the next number to sieve by once it hits the end of the list?

 

[Rahmuss]

I love this theorem: the Frivolous theorem of Arithmetic.

How true that theorem is.

I was thinking about some stuff and had some questions and thoughts.

1 - The distance between prime numbers seem to be mostly prime numbers themselves (although not always). Does this pattern hold that most are that way? Or maybe the higher the prime number goes, the more likely you'll have a non-prime number between them. Does anyone have a big list of primes with facts about things like this?

2 - Testing numbers by dividing them by primes. I'm sure there are better ways to check for prime numbers than just checking every single number. I figure numbers could move from one group to another to verify further and further whether they're prime or not. Any thoughts?

3 - Why are there prime pairs? What other interesting characteristics are there about primes? What do you mean by divisible by one and itself? Really what does that mean (deeper).

Is infinity a prime number? I know that sounds silly since infinity isn't really a number, and it's not really divisible by itself the same way that "regular" numbers are; but maybe prime numbers can be termed as some abstract quantity just as infinity is. I'm guessing this is pretty dumb; but just throwing thoughts out.

4 - The largest prime is (infinity/2) -1? Again... infinity... oh well. I'm not sure you could prove that there is a limit to prime numbers since you have that big old infinity there; but would there be a point where prime numbers would be in greater abundance as they get bigger? Like a wave function Cos(x). So at first you have a lot of primes... then the number drops rapidly... but could it go back up? Why or why not?

Please ignore my questions if they are too dumb.

 

[shmoe]

1. This is false, the distance between any two odd numbers is even, and the only even prime is two. Essentially the distance between two primes is primes when you have a pair of twin primes. (note this covers the distance between an odd prime and 2 being prime as well).

2. For a start, you only need to check for divisors less than a numbers square root. There's the Sieve of Erathosthenes, one of the simplest yet still effective ways to produce a list of primes. There are much better primality tests than thie naive divisibility test though, Elliptic Curve methods, AKS algorithm, and many many more, including specialized tests for numbers of a certain form (such as Mersenne primes).

3. Shucks, that's a tough one. In some sense the primes a pretty random, so it's not suprising if they sometimes get close to one another. No one knows yet if there are infinitely many of them, so it's not really possible to give a reason why they are there.

Being prime means you cannot be factored in anything but the most trivial way. You can think of primes as the building blocks of all natural numbers. Remember the Fundamental Theorem of Arithmetic?

Infinity is not a natural number, therefore it doesn't stand a chance at being called prime. First thing you have to do to be considered into the club of primeness is become a bonafide natural number, which infinity is really bad at doing.

4. There is no largest prime, there are infinitely many (you should be able to find the usual proof attributed to Euclid without much difficulty). The do become scarcer as you go out. If you were to grab some random number near a large number x, the probability that it's prime is about 1/log(x) (that's the natural logarithm). The prime number theorem says that is the number of primes less than or equal to x is about x/log(x).

However this isn't to say that there aren't quite a few of the little guys running around. You know $\sum_{n=1}^{\infty}1/n$ diverges but $\sum_{n=1}^{\infty}1/n^2$ converges, so in some sense there are far fewer perfect squares than there are natural numbers. Now $\sum_{p\ prime}1/p=1/2+1/3+1/5+1/7+1/11+\ldots$ diverges (the sum is over all primes), so there are still a considerable number in some sense.

 

[Rahmuss]

1. This is false, the distance between any two odd numbers is even, and the only even prime is two.

Oops! You're right... What I meant was the amount of numbers between them. i.e. between 7 and 11 there are three numbers (8, 9 and 10). Sorry about that.

2. For a start, you only need to check for divisors less than a numbers square root. There's the Sieve of Erathosthenes, one of the simplest yet still effective ways to produce a list of primes. There are much better primality tests than thie naive divisibility test though, Elliptic Curve methods, AKS algorithm, and many many more, including specialized tests for numbers of a certain form (such as Mersenne primes).

Wow! I had no idea. Could you use all of these tests? For example, use one test and all numbers that pass use the second test on them and so forth and see if there is a pattern or a quicker way to find primes? I'll have to look at some of those. Any discussions on PF for those?

3. Shucks, that's a tough one. In some sense the primes a pretty random, so it's not suprising if they sometimes get close to one another. No one knows yet if there are infinitely many of them, so it's not really possible to give a reason why they are there.

So this is more coincidental and can't really be reasoned mathematically?

Being prime means you cannot be factored in anything but the most trivial way. You can think of primes as the building blocks of all natural numbers. Remember the Fundamental Theorem of Arithmetic?

Primes are the building blocks in that they are more fundamental since they are only divisible by one and that number? What's the next step up? Having numbers divisible by 1, 2 and that number? What would those be called? Next divisible by 1, 2, 3 and that number? I guess in that sense there should be numbers that are more rare than prime numbers that would be more fundamental than prime numbers then. What would those be? What about numbers divided by fractions? I'm just rambling here, sorry.

Infinity is not a natural number, therefore it doesn't stand a chance at being called prime. First thing you have to do to be considered into the club of primeness is become a bonafide natural number, which infinity is really bad at doing.

LOL... yeah... sounds like infinity won't fit the picture very well. So in that sense you can't have an infinite amount of prime numbers since infinite is not a concept that would match a prime number sequence?

4. There is no largest prime...

I was just joking about that.

...there are infinitely many (you should be able to find the usual proof attributed to Euclid without much difficulty). The do become scarcer as you go out. If you were to grab some random number near a large number x, the probability that it's prime is about 1/log(x) (that's the natural logarithm). The prime number theorem says that is the number of primes less than or equal to x is about x/log(x).

Ok, that makes sense, even to someone as limited as I am in my knowledge of math and science.

However this isn't to say that there aren't quite a few of the little guys running around. You know $\sum_{n=1}^{\infty}1/n$ diverges but $\sum_{n=1}^{\infty}1/n^2$ converges, so in some sense there are far fewer perfect squares than there are natural numbers. Now $\sum_{p\ prime}1/p=1/2+1/3+1/5+1/7+1/11+\ldots$ diverges (the sum is over all primes), so there are still a considerable number in some sense.

Sum over all primes diverges... that sounds odd to me... would that mean that there are NOT an infinite amount of primes?

 

[shmoe]

1. Ahh, I see. This could be a function of looking at only small numbers where the distances between two primes is even smaller, and an unnaturally large percentage of "small" odd numbers are prime. Up to the first 100 or so primes the largest gap is 17 numbers long. 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.

2. If a test declares a number prime, why would you want to apply another primality test to it? There are some "probabilistic" primality tests where you do repeated applications, but I don't think that's what you meant. Google "primality testing", you'll probably find heaps of info.

3. It's probably not coincidental. There are convincing heuristic arguments (read-arguments with gaps in them) that would indicate there are infinitely many prime pairs. It is also known that there are infinitely many pairs p, p+2 where one is prime and the other has at most 2 prime factors, that's coming tantalizingly close to a solution (yet so far out reach still).

Next best thing to primes would be a low number of prime factors. In some ways they can be hard to separate (see paragraph above). Prime powers are also pretty decent too.

not sure what you mean about "infinite is not a concept that would match a prime number sequence", since you seem to agree there is no largest prime and therefore infinitely many of them.

4. Quite the opposite about the sum of recipricals of the primes diverging, showing this sum diverges is one way to prove there are infinitely many of them. If there were a finite amount, then this sum would have a finite value. However by adding up enough terms, you can make this sum as large as you like. It turns out if you add up 1/p for all primes less than say x, you'll get about log(log(x)), which although going to infinity very slowly, it still goes.

 

[Rahmuss]

1. Ahh, I see. This could be a function of looking at only small numbers where the distances between two primes is even smaller, and an unnaturally large percentage of "small" odd numbers are prime. Up to the first 100 or so primes the largest gap is 17 numbers long. 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.

I ran my program from about 1,000,973 to 1,002,361 and came up with these results:

7, 17, 3, 13, 5, 3, 13, 27, 11, 5, 1, 3, 13, 15, 29, 5, 13, 3, 13, 5, 21, 17, 29, 11, 11, 11, 7, 9, 1, 3, 19, 5, 15, 11, 5, 1, 11, 9, 19, 15, 11, 7, 23, 9, 25, 3, 17, 1, 11, 5, 17, 5, 27, 7, 9, 19, 9, 13, 3, 25, 9, 19, 39, 13, 3, 5, 1, 11, 9, 7, 71, 21, 7, 5, 5, 23, 3, 1, 5, 27, 31, 11, 11, 3, 5, 7, 9, 7, 11, 21, 5, 1, 21, 17, 35, 13, 5, 9, 1, 3, 25, 9, 41, 1, 3, 1, 9, 1, 15

The ratio here is presicely 27/109 (or about 1/4). Less than 2/3; but fairly close. Interesting that the ones that most come up which are not prime numbers are divisible by prime numbers... some big endless loop I'm sure, since you take those (ie. 9, 15, 21, 27) and we find that they are divisible by three and some other prime number... and it continues... I don't dare push my program too far. It already takes almost 0.10 seconds to calculate terms over 5,000,000. I did it in C++ if anyone wants the code (it's pretty small).

EDIT: Another quick note. The amount of numbers listed here always seems to be divisible by a prime number. Do they take that into account when they attempt to find a prime?

ADDITIONAL EDIT: They are divisible by any prime that I can calculate, except for the number 2 (of course), so I would submit that 2 is not in the same category... not that it couldn't still be a prime; but it's not the same type.

2. If a test declares a number prime, why would you want to apply another primality test to it? There are some "probabilistic" primality tests where you do repeated applications, but I don't think that's what you meant. Google "primality testing", you'll probably find heaps of info.

I did mean "probabilistic" there. You're right, it wouldn't make much sense using another prime number test on a prime number.

3. It's probably not coincidental. There are convincing heuristic arguments (read-arguments with gaps in them) that would indicate there are infinitely many prime pairs. It is also known that there are infinitely many pairs p, p+2 where one is prime and the other has at most 2 prime factors, that's coming tantalizingly close to a solution (yet so far out reach still).

Wow! Sounds like they're on the right track. I'm not great with number crunching. I just do the thinking.

Next best thing to primes would be a low number of prime factors. In some ways they can be hard to separate (see paragraph above). Prime powers are also pretty decent too.

Interesting. I wouldn't have thought of prime powers.

not sure what you mean about "infinite is not a concept that would match a prime number sequence", since you seem to agree there is no largest prime and therefore infinitely many of them.

I'm not sure either... I guess what I was getting at was that since infinity isn't really a defined quantity; but a concept, or limit, or... help me out here... anyway... since it's not a "natural number" (I think that's what you've called it), then saying that numbers (which happen to be "natural numbers") have an infinite limit (limit that is not "natural") seems to conflict. Though I'm sure it doesn't but it sounds like it.

4. Quite the opposite about the sum of recipricals of the primes diverging, showing this sum diverges is one way to prove there are infinitely many of them. If there were a finite amount, then this sum would have a finite value. However by adding up enough terms, you can make this sum as large as you like. It turns out if you add up 1/p for all primes less than say x, you'll get about log(log(x)), which although going to infinity very slowly, it still goes.

Interesting as well... I must have missed something in my math classes. I, for some reason, am not aware of the relationship between primes and log(x); but many people seem to have mentioned it. What is that relationship exactly?

 

[shmoe]

1. If a number isn't prime then it's divisible by a prime. Notice how small the gaps are (in your range 39 or less). The prime number theorem will tell us that the average gap sioze for primes less than x will be about log(x) (as there are x/log(x) primes). So if your looking at a bunch of "randomish" odd numbers that aren't much larger than log(x), you'd expect "many" primes. Furthurmore, the ones that aren't prime will have a prime factor no greater than their square root, so it's no surprise you see 3 often there. The only composite number less than 39 without 3 as a factor is 25 (which did occur in your range)

3. There's no problem with saying there are infinitely many primes. You accept that there are infinitely many integers? It's just a statement about cardinality of sets.

4. See the prime number theorem. The log there is really the cause of so many other logs. I'll have to think about a simple way of explaining why it makes an appearance there.

 

[Fredrick]

non-primes in prime positions, non-primes in non-prime positions

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).

 

[shmoe]

I don't understand what you think is new or worthy of a patent. If you've read this thread (or really any number theory text) you'd know that considering primes mod 6 is nothing new. Knocking off non-primes by considering all the numbers mod 1 or 5 mod 6 then removing multiples of 5, 7, etc. is just a slightly modified sieve of erathosthenes. This pattern of "jumps" in lines when you're removing multiples of a prime p is not difficult to explain, you're just adding this prime again until you hit 1 or 5 mod. If p=6k+1 then your "link pattern" is 4k, p-4k. If p=6k+5 it's 2k+1, p-2k-1 (both these are easy to prove).

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 misleading as this pattern does not continue if you're putting primes in the left column, it's only there because there are no composites congruent to 1 or 5 mod 6 less than 24. Though it does occur to me that you might want to include 25 on this table as well (and all numbers congruent to 1 or 5 mod 6), in which case this 'link-pattern' is just a trivial observation (simple to prove with modular arithmetic) that says nothing about primes.

Including 0 as a natural number or not is totally irrelevant to how any number theorist will think. It's just a convention that's been adopted over the years as being the most convenient (or at least not terribly inconvenient). Number theorists are not at all restricted to working with the natural numbers and will jump around to integers, algebraic number fields, complex numbers, and more exotic systems, and I assure you never forget about the existence of 0.

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. For this reason it behaves sufficiently different from 2, 3, 5, 7, ... etc. that it doesn't share most of their interesting properties. Of course you could include 1 as a prime if you really wanted to set your definitions up that way, but you'd find many theorems beginning "let p be a prime other than 1". It's just a definition made to simplify discussions that's generally been agreed upon.

 

[Rahmuss]

It seems throught the test that I've run, that:

All prime numbers have $$n$$ numbers between them.
Where $$n$$ is the product of two prime numbers.

This works with all prime numbers except for "2". 2 is never used to find $$n$$ and between 2 and 3 there are 0 numbers, and no two primes multiplied together make 0. Any thoughts on that?

Could there be a function to find all prime numbers where $$n$$ is 3? Or a function where $$n$$ is 5, or 7, or any other prime number? Where the function would pass through each of the prime numbers (either the first in the pair of the sequence or the last in that pair).

 

[shmoe]

All prime numbers have $$n$$ numbers between them.
Where $$n$$ is the product of two prime numbers.

If you go out far enough, you'll see this is false, the next prime after 1098847 is 1098847+106, there are 3*5*7 numbers between them. I mentioned the conjecture that there are infinitely many twin primes, Hardy and Littlewood gave a more refined conjecture that gives an asymptiotic for how many their are (like the prime number theorem). It goes beyond this to predict asymptotically how often any size gap will appear (recall twin primes correspond to a gap of 1). Long story short, all possible gaps are expected to appear infinitely many times (though not all at the same frequency).

Could there be a function to find all prime numbers where $$n$$ is 3? Or a function where $$n$$ is 5, or 7, or any other prime number? Where the function would pass through each of the prime numbers (either the first in the pair of the sequence or the last in that pair).

Sure, there can be a function that does this, but it's almost surely not what you are hoping for. Let f(k)=kth prime p that is followed by n composites then a prime. Not at all computationally useful, but it's cetainly a function. Probably for any simple restrictions on what you're hoping this function to satisfy, it will either be impossible or at least not possible by todays technology.

 

[matt grime]

Of course between any pair of twin primes there is 1 number and 1 is not the product of two primes.

 

[shmoe]

Of course between any pair of twin primes there is 1 number and 1 is not the product of two primes.

Good point- for some reason I assumed he meant at most two primes.

 

[Rahmuss]

Of course between any pair of twin primes there is 1 number and 1 is not the product of two primes.

Oh, you're correct. I was considering "1" as a prime and "2" as a non-prime. Although it's not really the "Prime" number system if done that way. So it's a different number system I guess that has the same numbers except for 1 is included, and 2 is excluded. Other than that it works.

Yeah... I guess the only way to get a true function for the infinite series is to know all values of the series (or what they will be), so the function given won't help out. We need a function that would give answers by plug and play.

There's got to be another pattern... come on, you guys are math wizards... think, THINK!

Do you think it can actually be solved? Or do you think there really is no pattern so it can't really be solved?

 

[Rahmuss]

So how about this:

All prime numbers have $$n$$ numbers between them.
Where $$n$$ is a product of primes.

Does that work? (again with the exception of "2").

 

[shmoe]

So how about this:

All prime numbers have $$n$$ numbers between them.
Where $$n$$ is a product of primes.

Does that work? (again with the exception of "2").

This is trivially true (barring the "1" exception, unless you conisder an 'empty product' a product). Remember the Fundamental Theorem of Arithmetic?

 

[Rahmuss]

...Remember the Fundamental Theorem of Arithmetic?

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; 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.

Funny that the theorem was stated by "Hardy and Wright" and my last name is Wright and my cousins last name is Harding (close enought).

I still think there is a clue in what the $$n$$ is that will give useful information.

So will it be solved then? So far, all we have are ideas which give only few prime numbers, or ideas which give probable prime numbers. No plug and play yet.

 

[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".