Building
up the Model
With
a similar reasoning we ask ourselves, how many prime numbers
exist by ranks of number of digits, under the assumption
of order in the "positional notation" or "the
relative value of the figures", developed by the hindu
(beginning of VI century BC) and recognized as one of the
major advances of mathematics, as the science of measurement
and order.
The
formal procedure developed is:
Definition:
If
 |
| |
where: |
|
:
:
|
 |
| in
short the formula becomes: |
 |
|
| |
| Convergence:
the series is convergent because |
 |
In
other words exist
|
|
| Monotonicity:
|
|
| Which
yields a rapid asymptotic curve towards zero (see figure
1) |
| Scarcity: |
|
|
|
The empirical procedure developed under this method shows
the following brief table by calculating the sum of the reciprocals
in numerical figures:
|
n
|
Dig
|
1/dig
|
CRAA
- 2
|
|
0
|
4
|
0,250000000000000
|
|
|
1
|
21
|
0,047619047619048
|
0,297619047619048
|
|
2
|
143
|
0,006993006993007
|
0,304612054612055
|
|
3
|
1061
|
0,000942507068803
|
0,305554561680858
|
|
4
|
8363
|
0,000119554315437
|
0,305674135996295
|
|
5
|
68906
|
0,000014512524308
|
0,305688648520603
|
|
6
|
586081
|
0,000001706248795
|
0,305690354769398
|
|
7
|
5096876
|
0,000000196198613
|
0,305690550968011
|
|
8
|
45086079
|
0,000000022179795
|
0,305690573147806
|
|
9
|
404204977
|
0,000000002473992
|
0,305690575621798
|
|
10
|
3663002302
|
0,000000000273000
|
0,305690575894798
|
|
11
|
33489857205
|
0,000000000029860
|
0,305690575924658
|
|
12
|
308457624821
|
0,000000000003242
|
0,305690575927900
|
|
13
|
2858876213963
|
0,000000000000349
|
0,305690575928292
|
The result of the convergent series is a constant, a constant
named CRAA-2 (see THE MATHEMATICAL CONSTANT OF AGUILAR-ACHA,
Ciencia y Computación, EL DIARIO, 22/VII/99, La Paz-Bolivia).
Numerical
Result
This new
Aguilar-Acha's constant, CRAA-2, is originated in the former
set of: a) a formal definition, b) expressions and formulas
and c) construction of the table up to 1013 decimal digits
and of the primes comprised within each rank, associated to
the advancement of the study of the theory of numbers, which
represents to be the infinite value:
|
CRAA
- 2 = 0,30569057592 …
|
CONCLUSION
The constant
CRAA-2 is thus obtained and preliminary verifyied. In measurement
terms this result suggests and help us in the decoding of
primes' laws or rules related to its important regularities
and is worthy of our attention concerning the outstanding
and most important prime numbers, as objects or autonomous
elements, in appearance just refractory to any kind of relations
or regularities.
APPLICATIONS
Remembering
that a beautiful theorem states that: "all numbers can
be written as a product of primes", already demonstrated
by other mathematicians, the constant CRAA-2 help us to prove
the conjecture of scarcity of the primes in the development
of its series and the density or distribution of primes. Which,
undoubtedly, is of great importance for the study of the shape
and construction of the structure of the mathematical science.
Also,
in applied mathematics, to build algorithms for the calculation
of very long figures and the processing of chains, with precision
and high security in computing machines with high capacity
and speed, to solve many complex problems, perform investigations,
notable classifications and hierarchizations, etc. useful
to judge numerical propositions on the theory of numbers and
other branches of pure and applied mathematics and related
sciences.
REFERENCES
R. Aguilar-Acha, La Constante de Aguilar-Acha, Ciencia y Computación,
EL DIARIO, La Paz-Bolivia.
R. Aguilar-Acha, Decoding the Secrets of the Law of Primes
in the Theory of Numbers, www.bolivialinux.com
(2001)
G.H. Hardy and E.M. Wright, An Introduction to the Theory
of Numbers, Oxford Science Publications, (1979)
W.J. LeVeque,Fundamentals of Number Theory, New York, Dover,
(1996)
Further
information: cellular 775-22299 web site www.telecel.com.bo,
e-mail: raguilar40@starmedia.com
or raguilar40@terra.com
or telephone (591-2)-2485559, La Paz-Bolivia, South America
(All rights reserved).
L.P. 12/X/01.
|
