**
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__ G_{n} = P_{n}
– P_{n -1 } & __postgap__
g_{n} = P_{n+1} – P_{n}

g_{n} º
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__

V_{n}
= g_{ n }– G_{n}

( V_{n} = G_{ n+1 }–
G_{n })

V_{n}
=**
**
(P_{n+1}**
**
–**
**
P_{n}) –**
**
(P_{n}**
**
– P_{n-1}) = P_{n+1}**
**
–**
**
2P_{n}+**
**
P_{n-1}

*
*

In*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 P_{n}^{’}
and variation as P_{n}^{”}.

Both gaps (pregaps and
postgaps) and variations

involve 3 primes: P_{n-1}, P_{n},
and P_{n+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 ]
__