Not knowing patterns does not mean that patterns don’t exist. It just means that we are not intelligent enough to discover the patterns. Example: I keep reading that the non-trivial zeros of zeta in the critical line are random and unrelated, when there is a clear algebraic expression that links any two NT zeros and the Harmonic function.

]]>je vais bien merci et vous

]]>The set of prime numbers can be expressed as (Caceres, 2017):

{Primes} =

{2,3}

∪

{6kn+1 | kn≠6xy+x+y and kn≠6xy-x-y for all x,y∈N}

∪

{6km-1 | km≠6xy-x+y for all x,y∈N}

The generation of primes using this algorithm is complete based on the following observation:

1: With k=6xy+x+y, we have 6k+1 = 36xy+6x+6y+1 = (6x+1)(6y+1), i.e. all products of two factors both equivalent to +1 (mod 6);

2: With k=6xy-x-y, we have 6k+1 = 36xy-6x-6y+1 = (6x-1)(6y-1), i.e. all products of two factors both equivalent to -1 (mod 6);

3: With k=6xy-x+y, we have 6k-1 = 36xy-6x+6y-1 = (6x+1)(6y-1), i.e. all products of two factors, one equivalent to +1 (mod 6) and the other equivalent to -1 (mod 6).

Starting with the integers equivalent to ±1 (mod 6) and excluding these three sets leaves those integers equivalent to ±1 (mod 6) which cannot be represented as a product of two factors equivalent to ±1 (mod 6), i.e. the primes p≥5.

The first numbers in the generator series kn:

kn= 1, 2, 3, 5, 7, 10, 11, 12, 13, 16, 17, 18, 21, …

Generating primes pn=6*kn+1:

pn= 7, 13, 19, 31, 43, 61, 67, 73, 79, 97, 103, 109, 127,…

The first numbers in the generator series km:

km= 1, 2, 3, 5, 7, 8, 9, 10, 12, 14, 15, 17, 18, 19, 22 …

Generating primes pm=6*km-1:

pm= 5, 11, 17, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131…

]]>For 16 years now I have been on the track of the Collatz conjecture, and whilst I know the experts have been unable to solve it since 1937 using traditional methods, I have been employing some alternative methods and am finding distinct patterns emerge. I believe there is a solution to that conjecture as well and someone one day will solve it, maybe not me but there is a pattern there, it only has to be coded.

]]>No, it isn’t. I am asking about the formal conjecture, which can (in principle) be proved, and, after this, all these `heuristic motivation’ would become a strict proof. There is a formal definition of a random sequence ( see e.g.

http://mathdl.maa.org/images/upload_library/22/Ford/Volchan46-63.pdf ), and, for any random sequence A, all these “conjectures” (like there are infinitely many members p,q of A with |p-q|=2) are very easy theorems. This, however, is the formal definition of what is called “Basic heuristic” at https://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ Because primes is not completely random sequence, we need to formalise the notion of “Advanced heuristic” to formulate the iniversal conjecture…

I don’t think there is a formal one-size-fits-all conjecture for this, but I expand a little more on this heuristic at https://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/

]]>Cramér’s model, providing heuristic motivation for numerous conjectures, is pretty much it.

]]>pattern beyond the obvious ones”. Are there exist a formal conjecture which formalises this statement? This would be a “grand unifying conjecture” of the prime distribution theory, which would imply the prime tuples conjecture (including twin prime conjecture), the Riemann Hypothesis, the Cramér’s conjecture, and all other not-yet-asked questions which are easy to verify assuming that “the primes do not observe any significant pattern beyond the obvious ones”.

If not, what do you think is the major difficulty in the formulation of such a conjecture?

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