the last days have been surprisingly fruitful and have reinforced my

conviction

that it is quite easy to prove the Twin Primes Conjecture and de

Polignac's using just algorithmic number theory.

I published an article in 2009 that contains many of the ideas I'll

talk about here

http://www.unixgarden.com/index.php/gnu-linux-magazine/un-algorithme-additif-et-iteratif-pour-construire-les-nombres-premiers

There, I show good signs that the TPC is true but there are several

shortcomings,

for example nothing protected me from a "number conspiration".

I think have solved them now, thanks to new insights.

The TPC, Cousin and Sexy primes all appear as special cases derived

from the

more general de Polignac conjecture, which I believe is now in my

reach,

but I won't go this far right now.

Refresher :

-----------

I study "First differences of reduced residue systems modulo primorial

numbers"

that are familiar to those who deal with sieves and wheel sieves in

particular. For short, since

these are cyclic, I call them wheels, it's more convenient. The first

wheels are :

W1 = [ 1 ]

W2 = [ 2 ]

W3 = [ 4, 2 ]

W5 = [ 6, 4, 2, 4, 2, 4, 6, 2 ]; => OEIS A145011 "First differences of

A007775." (repeated infinitely)

W7 = [ 10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6,

6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4,

2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6,

4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2 ] => A049296 "First

differences of A008364.

W11 is way too long for this post and the size only gets worse but you

get the idea.

The numbers in these wheels are the gaps between consecutive prime

number candidates.

You still have to sieve out multiples of larger primes. The best

analogy is a gear with

p# teeth but only some of them are kept because the others are

multiples of p and below.

* Spinning W1 would generate the whole set of N

* W2 generates the set of odd numbers (2n+3)

* W3 creates the numbers 6n+/-1

Imagine that if you got the wheel of infinity, you then get all the

gaps between primes.

Here are some more or less known properties :

(More properties are welcome, if you know them)

* the sum of all the numbers in the wheel of prime p is p# (primorial)

* the number of elements in a wheel is [ 1, 2, 8, 48, 480, 5760,

92160, 1658880, 36495360 ...]

=> OEIS A005867 "Local minima of Euler's phi function"

(a bit like p# but you multiply all primes-1 instead)

* OEIS A049296 also says they are palindromic (there is one axis of

symmetry)

* Very interesting for the TPC : starting with W2, the last gap is

always 2

(still OEIS A005867) => see later

* easy to prove (see later) : starting with W3 the other symmetry

point is 4

(From here we have good hints that the twin primes and cousin primes

conjectures come on a silver plate)

* Even more interesting, still quoting OEIS :

"The palindromic part starts and ends with p_(n+1)-1 for the n-th

primorial number."

So the first element is an almost direct indication of the next

prime.

* It also means that, starting with W3, the second element will be the

actual gap between the next prime and the following.

Algorithmics :

--------------

The study of the infinite set of primes requires

an infinite wheel that is not possible to study (well, hmmm, acurately

at least)

but we can know what will happen there by studying the transformations

of

the wheel Wp(n) to create Wp(n+1). This is where algorithms show up.

In http://ygdes.com/~whygee/srs.html I have merged my algorithmic

approach

with the remarks made by Steve Maddox on this list

http://tech.groups.yahoo.com/group/primenumbers/message/21637

It's a 3-steps algorithm that creates a new wheel W2 from any given

valid wheel W.

* generate => create the new prime p2 = W[1]+1

* replicate => create W2 = concatenate p2 copies of W

* merge => remove pairs of gaps that would generate multiples of p2 :

length(W) times : add a pair of gaps and shrink W2 by one gap.

The merge part is the most interesting part and is the key to the

proofs

so it is important to understand its properties.

* One simple method used in 2009 (and commonly used when manipulating

sieves)

is to make a congruence test on the sum of the gaps. When the

accumulator

is zero, merge the offending gaps. But its insight level is close to

the sieve of Eratosthenes : it tells what primes are not (composite)

and

not what they are.

* Steve has shown that the gaps between W2's gaps is derived from W.

Using the same accumulate & test algorithm as before, an algorithm

can merge pairs when their distance is p*W[i] (for successive indexes

i)

These algorithms have allowed me to explore wheels that I couldn't

study with pen&paper only, and validate my approach.

These are not the most efficient algorithms ever but look at the source

code

of the example at http://ygdes.com/~whygee/srs.html : it's quite

simple

and easy to analyse. For example :

* Since the positions of the merged pairs are determined by W, which is

symmetric,

W2 is symmetric too. (Or palindromic if you prefer)

* It follows that W[length(W)]=2 and W[length(W)/2]=4 are two

symmetry points of the wheels (starting with W3)

and no change is to be expected (unless you generate composites).

* BTW, creating W2 requires length(W) merges, one important data

to keep in mind for later.

Analysis of the W->W2 transforms

--------------------------------

Proving the TPC is equivalent to proving that there is an infinity

of wheels where the second number W[2] is equal to 2.

Proving de Polignac is the same but with W[2]=2n.

So we have to study how this second number is created :

it comes from the following gaps, that get shifted

toward the beginning during each "merge" operation

after each successive transform.

At this point the proofs split into two main branches,

and I have used a producer-consumer approach :

* Producer : the replicate step.

* Consumer : the merge step

- The first branch of the proof is the persistence of gaps :

prove that there are more gaps produced than consumed.

It is not difficult to prove that the wheels

always contain at least one gap once it is created by a merge.

More generally : the algorithms help to quantify/characterise

the population of a given wheel.

If one gap disappears, it might not shift down to position 2 anymore.

This is critical for the gap 2 that is not created after W2.

Gap=1 disappears almost instantly...

The 2009 article also contains a little proof (by absurd) that the gap

2

can not disappear from the wheels. I don't know why I limited this

proof

to the gap 2, it works very nicely for any gap.

This is confirmed by the properties that I have mentioned above :

there is always at least a gap=2 and a gap=4 starting with W3.

Since each transform creates length(W)-2 copies of EACH gap of W

(not counting the length(W) new gaps created by the merges)

it is not possible that any gap disappears from any W

(except gap=1 in W1).

Using a less insightful algo in 2009, I have run calculations

that showed that the number of gaps=2 grows nicely, corresponding

to the series 1*3*5*9*11*15*17*21*...*(p-2)

Even though the proportion of gap=2 in a wheel tends toward 0,

the actual quantity actually increases. And it's only considering the

TPC.

So I consider this first branch valid.

- The second branch of the proofs is the "survival" of a given gap,

as it slowly shifts towards W2[2] to become a prime gap, avoiding

merges.

There is the eventuality that some gaps get replicated but never reach

W2[2]...

It is less straight-forward but a few days ago, I found new formulas

that give the values of the merged gaps (in effect removed from the

concatenated wheel) and where (without using accumulation or

congruence).

From there it is possible to predict which gaps get merged after x

transforms.

Thanks for reading so far

-------------------------

Sorry for remaining vague at this point but as the title says, it's

just an outline of a work in progress

and I'm currently writing a longer, better, detailed report on this,

with examples, working code etc.

I have no affiliation so I don't see how I could post anything to

arxiv. djb told me

to post here anyway for timestamping/anteriority and for the eventual

"hope that someone replies".

I'm sure that de Polignac and his contemporaries 2 centuries ago were

well aware of

many things I just wrote here so why couldn't they prove the TPC ?

There are well understood properties and structures,

and so far the "conspiracy of the numbers" has only led me to a

well-ordered,

predictible system, whose byproduct is the series of random-looking

prime numbers.

Feel free to comment constructively.

I don't post to troll but to identify potential mathematical flaws.

For example, is there another branch to consider to strengthen the

proofs ?

Have a nice day,

Yann Guidon