Facinating experiment with primes and resulting theory

[thetexan]

I had an idea a while back and did an experiment concerning primes.

It began with the idea that the distribution of prime numbers must be somehow determinable.

So I used photoshop and created a white line several pixels wide and several million pixels long, each pixel along the length representing a positive number. Then, using the pattern capability of photoshop, I changed every other pixel along the length to black representing a seive that eliminates all multiples of two. Then I repeated it for three and so on. After doing it with two I had a pattern of every other pixel being black. After each iteration a new repeating pattern would appear. The pattern was repeating and predictable. Of course with each pass of the sieve a much more complex pattern emerged.

Now, there are a few observations with this.

1. The pattern represents the pattern of numbers removed from the number line.
2. The inverse pattern i.e. the pattern of the numbers left is much more complex but a pattern none-the-less.
3. As the number of passes of the sieve increases the pattern of removed numbers increases greatly in complexity and that pattern's inverse is even more so.
4. No matter how complex the pattern, we can deduce that there is a pattern and therefore also deduce that there is an inverse pattern.

5. And if there is a pattern, it must be definable by some algorithm, no matter how complex.

Doesnt this idea and experiment indicate that it should be possible to exactly determine and predict the pattern and frequency of primes?

We know there is a pattern...patterns are definable...therefore prime number distribution must be definable.

What do you think?

 

[epsi00]

I like your idea. Can you post some pictures to look at. Maybe the primes patterns are fractal ( remember Ken Ono ).

 

[coolul007]

ah! welcome to the law of small numbers, if there is a pattern it would predict this gigantic prime 10 billion digits long. But predicting Primes is like proving a negative proposition.

 

[thetexan]

there IS a pattern. I tried it up to number 11. After applying the sieve to the number line up thru 11 the pattern repeated itself about every 3500 numbers. After 7 it was only repeated about every 200 numbers. It makes intuitive sense. Of course there is a pattern. After 13 it will have a repeat of who knows how long.

But it raises the following questions.

We can deduce that there will be an ever increasingly complex pattern as the numbers in the sieve increases. Astronomically so! But a pattern none the less. Since these patterns are make up of the 'overlaying' of each pattern for each number as it is laid down on the numberline, shouldn't there be a formula that can be derived that describes that pattern?

Since this pattern represents the numbers that have been removed from the set of all numbers as each pass of the sieve does its work, what remains is the anti-pattern of the numbers left in the set from which the next and all remaining primes will be found. The 'next' prime is always the lowest number in this remaining set. If a formula for the pattern can be found (in theory) why can't a formula for the anti-pattern be found since it itself is a pattern?

This will all be very complicated but doesn't this mean that there is a definable, predictable formula for determining the next prime?

tex

 

[lurflurf]

There are patterns and formula. When people say things like "There is no formula for prime numbers and no patterns." they are speaking lossely. What is they mean to say is any patterns or formulae that are known or easily found are not usefull. Any formulae or paterns are of comparable complexity to the primes themselves. In other words it would be a big deal to find an easier way to find prime numbers. If right now we can take 10000$ in computer hardware and 1000 hours and do some task (factor a n digit number or find m primes ect.) we would like to find a method where we can take 9800$ in computer hardware and 970 hours and do the same task.

 

[coolul007]

I would love for there to be a pattern, but my argument against it is that NOT 2n does not tell me anything about the numbers in the NOT set. Much like NOT a horse doesn't tell me about the diversity of biology.

 

[thetexan]

Well, I thought about that also. Just finding a formula to that pattern may be just as complicated in itself as finding one for primes. My only thought was that I have often heard that the SEEMING randomness of the distribution of primes makes it difficult to make any predictions. My experiment, at least to me, in my fairly elementary knowledge of math (especially compared to the guys around here) showed me that the distribution is in fact ordered and predictable, at least as far as I took it, theoretically. And even though I only went up to and confirmed the pattern up to 11 (because the repeat soared to over 3000) it showed that there is a pattern.

The main idea I had was that the formula for the sum of those patterns (in effect, the overlaying of those patterns, one over the other) might be something like the sum of a series. The anti-pattern which represents the set of remaining numbers, from which the next prime will be the the lowest valued number, would also be the sum of a different series, which could be determined after having solved the first series.

Again, the complexity of this itself might make it just as hard as any other way. I just thought the idea of a series might be a novel approach that hasnt been tried yet.

tex

 

[coolul007]

I have thought about this also, as every time you run into a twin prime, the formula breaks down. There may be an infinite series of "generators" for primes, much like Sophie Germain primes extended until it fails. 2(2)+1,2(5)+1,2(11)+1,2(23)+1, 2(47)+1=failure. It seems that some nth iteration of a formula is always divisible by n.

 

[thetexan]

coolu007,

Im not so sure about that. For example. Starting with 2, remove all multiples of 2 and place them in a set. I now have two sets. All of the multiples of 2 and a set with everything else, which happens to have a pattern, all of the odd numbers. Take the lowest number of these and you have the next prime...3. Now remove all of the threes from the set and the those numbers now have a new pattern. The remaining numbers also have the inverse pattern. Take the lowest number in that set and you have the next prime, 5.

There must be some relationship between the pattern, the inverse pattern and the next prime in the chain.

 

[coolul007]

Their are two ways to create a series, using the previous term(s) to create the present one or a general formula. Most series, can be discovered to do both. Such as the Fibonacci series where the present term is the sum of the 2 previous terms. There is a formula for the nth term. discovering it took quite a while, but it was done. It is a Binet formula. The sieve method(s) can't give me the nth term, only the next term. A prime formula is a formula that will give me any term I want immediately without generating all the prime prior to it in sequence. or without knowledge that there are other primes, I.E the 10th perfect square is 100.