the last days have been surprisingly fruitful and have reinforced my
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
There, I show good signs that the TPC is true but there are several
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
more general de Polignac conjecture, which I believe is now in my
but I won't go this far right now.
I study "First differences of reduced residue systems modulo primorial
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
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
* Very interesting for the TPC : starting with W2, the last gap is
(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
So the first element is an almost direct indication of the next
* It also means that, starting with W3, the second element will be the
actual gap between the next prime and the following.
The study of the infinite set of primes requires
an infinite wheel that is not possible to study (well, hmmm, acurately
but we can know what will happen there by studying the transformations
the wheel Wp(n) to create Wp(n+1). This is where algorithms show up.
I have merged my algorithmic
with the remarks made by Steve Maddox on this list
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
* 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
so it is important to understand its properties.
* One simple method used in 2009 (and commonly used when manipulating
is to make a congruence test on the sum of the gaps. When the
is zero, merge the offending gaps. But its insight level is close to
the sieve of Eratosthenes : it tells what primes are not (composite)
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
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
of the example at http://ygdes.com/~whygee/srs.html
: it's quite
and easy to analyse. For example :
* Since the positions of the merged pairs are determined by W, which is
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 is equal to 2.
Proving de Polignac is the same but with W=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
can not disappear from the wheels. I don't know why I limited this
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
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 to become a prime gap, avoiding
There is the eventuality that some gaps get replicated but never reach
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
From there it is possible to predict which gaps get merged after x
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
predictible system, whose byproduct is the series of random-looking
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
Have a nice day,