fun with primorials and prime numbers

Hi,

If you want to contribute something more useful, you could help me find this sequence of pisano periods:

ie:

I'm looking for that up to 10million+ length to check for patterns, as apparently it is unknown if there is a pattern in it.

cheers, Jamie

But what do you mean by random distribution?

Dirichlet's theorem applies to both.

You wrote "I used the term pseudorandom interchangeably with random" a couple of posts back. Your words, not mine. But if you understand the difference between them (or at least understand that there /is/ a difference), then we should put past mistakes behind and go forward.

You should not assume the result you are trying to find at the start of your experiments (especially when mainstream mathematics suggests that your hypothesis is wrong). You can /hope/ that your hypothesis is right, but you don't assume it - you keep an open mind. Perhaps you will find a way to prove that there is no formula for generating primes, and their distribution truly is random - mathematicians would love a proof for that.

I believe I /have/ been contributing usefully - I have been trying to encourage you towards logical and critical thinking, accurate use of terminology, and a scientific viewpoint.

If you prefer, I could usefully give my blunt opinion on where this is all going, but I would much rather be optimistic and encourage your curiosity.

Hi,

It is quite the tangent, discussing whether the primes are random or pseudorandomly distributed, as my original question that mentioned pseudorandom was this:

"I think the count of primes in the two formulas has to be defined as equal, if they weren't equal then how can the primes have a pseudorandom distribution?"

That question is the same whether the word random or pseudo random is used, so it is really an irrelevant tangent to try to figure out random or pseudorandom distribution of the primes in that context.

cheers, Jamie

Hi,

I appreciate your candor, and I shouldn't believe that the primes have a pseudorandom distribution obviously, but when I stated:

"I think the count of primes in the two formulas has to be defined as equal, if they weren't equal then how can the primes have a pseudorandom distribution?"

Do you see how by "begging the question" here that the primes have a pseudorandom distribution, I am saying that even if the primes have a pseudorandom distribution, the counts of the two formulas should still be the same, and of course if the primes have a random distribution the counts of the two formulas would be the same too. In either case the count of primes in the two formulas should be equal I think.

I am not (and was not) trying to prove that the primes have a pseudorandom distribution from my original question.

cheers, Jamie

Hi,

I just made a cool discovery that 6n+1 and 6n+5 have some periodic (non random) patterns of their relative prime distributions.

For 6n+1 for the non negative integers, all the primes in it, ie starting with:

6n+5 produces primes:

Now I make a new binary sequence where 6n+1 primes are assigned 0 and

1 (5 = 6n+5 prime so binary1)

Then randomly today I came across this news story about something very similar about prime distributions being biased in adjacent residue classes:

The formulas I am using, ie 6n+1 and 6n+5 are reduced residue systems I think..

As someone on sci.math replied to me a couple weeks ago:

"That list of 5760 numbers which have no common factor with 30030 is called its "reduced residue set". It contains 1, all the primes from

So.. from all that it makes sense that some non-random periodic patterns should be visible when comparing 6n+1 and 6n+5 "reduced residue sets". The way I have done this is by combining the two sets of primes from 6n+1 and 6n+5, and assigning a 0 to 6n+1 primes, and a 1 to 6n+5 primes, and making this into a new binary sequence, then checking that sequence for periodic patterns (comparing it against a pseudorandom binary sequence).

I did it for 1million sequence length, and am currently running the code for 3million sequence length to see, but there are slightly more periodic patterns in the 1million sequence length for the 6n+1/6n+5 compared to the pseudorandom binary sequence.

What I really want to do is a "higher" reside system, ie y=30x+b (30 primorial) where there are 11 b's that for all non negative integers will create more than one prime y.

And then using base11 (or a subset of the 11 b's ie base10) create a new sequence like the binary sequence for 6n+1/6n+5, except a base10 sequence, and then see if that has many periodic patterns compared to a pseudorandom sequence, also permutations of the reduced residue sets, ie maybe 3 or 5 of the b's etc turned into the single sequence.

So I think there are self referential patterns in the primes and probably patterns in the primes related to things outside the primes too, just gotta find them..

But a good quote about this on John Baez blog:

" If you have a prime that ends in a 7, what's the probability that the next prime ends in a 7?

I would still expect the answer to be close to 25%. But these mathematicians, Robert Oliver and Kannan Soundarajan, actually looked!

And they found that among the first hundred million primes, the answer is just 17.757%. That's way off! ...

Of course, what's really surprising is not this correlation between the last digits of consecutive primes, but that number theorists hadn't thought to look for it until now! "

So it's always good to look for patterns even if they seem simple especially if you have a computer it is easy to do.

I didn't find the pattern in this news story, but I think there is a more general pattern of primorial reduced residue sets that is interesting to try to figure out (probably explained already by the prime k-tuples conjecture as stated in the quantamagazine.org article linked above.

(basically I have no idea what I'm doing but it is fun!)

cheers, Jamie