Introduction of Variation

Here the sequence of primes is presented graphically in terms of a step function or counting function which is traditionally denoted p(x). (Note: this has nothing to do with the value p=3.14159...)

Dynamic structure of allocation of prime numbers 
in natural numbers series
is constituted by

Gaps between consecutive primes 
Pregap Gn = Pn Pn -1   &    postgap gn = Pn+1 Pn

gn G n+1

n ordinal number of a prime

After  creating the chart of Gn  (G=f(n)) ,

we see immediately in the ocean of the apparent chaos

Islands of some regularities:






Symmetrical forms,



Symmetrical each other forms,



And repeating forms






These forms are constituted not by gaps themselves,
But by
differences of consecutive gaps -
Variations of gaps

Vn = g n Gn

( Vn = G n+1 Gn )

 Vn = (Pn+1 Pn) (Pn Pn-1) =  Pn+1 2Pn+ Pn-1

troduction of gaps has been the first step 
of prime numbers dynamic structure - 
introduction of
VARIATIONS is the next, more profound, one
creating a great advantage in research of prime numbers.
It is proved to be very effective 
by having discovered many unexpected regularities.

There may be some remote analogy
between the dynamic structure 
of discrete array of prime numbers and calculus 
if consider gap as Pn and variation as Pn.

Both gaps (pregaps and postgaps) and variations 
involve 3 primes: Pn-1, Pn, and Pn+1.


Up ] [ Introduction of Variation ] Regular and Repeating Groups ] Groups in the Range until 100,000 ] Groups in the Range until 1,000,000 ] Key of Primes Structure ] Distribution of gaps and variations ] Boolean Algebra of Classes ] Consecutive primes ] Groups of primes in the infinite set ] Groups of primes in a limited range ] We have for each kind of regular and repeating groups ]


Last updated 03/27/2009
Copyright 2003 Michael Chassis. All rights reserved.