Pi and Golden Number: not
random occurrences of the ten digits. Jean-Yves
BOULAY Independent
researcher - 43 rue des Albatros
72000 LE MANS - FRANCE jean-yvesboulay@orange.fr |
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Abstract: This paper demonstrates that the order of first
appearance of the ten digits of the decimal system in the two most fundamental
mathematical constants such as the number Pi and the Golden Number is not
random but part of a arithmetical logic. This arithmetical logic is identical to Pi to its
inverse and to the Golden Number. The same arithmetical phenomenon also
operates in many other constants whose square roots of numbers 2, 3 and 5, the
first three prime numbers.
1.
Introduction.
The number Pi (π) and the Golden
Number (φ) and the inverse of these numbers are made up of a seemingly random
digits. This article is about order of the first appearance of the ten figures of
the decimal system in these fundamental numbers of mathematics. There turns out that the
ten digits decimal system (combined here with their respective numbers: figure
1 = number 1, figure 2 = number 2, etc..) do not appear randomly in the digits
sequence of Pi (π) and the digits sequence of Golden Number (φ). The same
phenomenon is also observed for the inverse of these two numbers (1/π et 1/φ).
1.1. Method.
This article studies the order
of the first appearance of the ten figures of the decimal system in the
decimals of constants (or numbers). After location of these ten digits merged then
in numbers (figure 1 = number 1, etc), an arithmetical study of these is
introduced.
2.
The Ratio 3/2.
The sum of the ten figures of
the decimal system, considered as numbers in this article, is 45:
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7
+ 8 + 9 = 45
This number is sum of two others: 45 = 27 + 18. These two numbers have a
ratio to 3/2 and are respectively equal for 3 times and twice 9. The number 10,
which here represents the ten possible occurrence ranks of the ten figures of
decimal system, has the same characteristics: sum of two other one numbers with
a ratio to 3/2:
10 = 6 + 4
2.1.
The Ratio 3/2 inside constants π and φ.
Figure
2 analyses the constant Pi (π). In this table, the ten digits of the decimal system are
identified (a) and ranked in order of their first appearance (c). At last
arithmetical analysis is presented: the sum of the first six values and the
last four in a ratio to 3/2. All tables in this article use the same type of
set-up with an arithmetical area (d) more or less developed.
It appears that for Pi, the
ten digits of the decimal system are organized in a ratio to 3/2: the sum of first six digits is to 27 and the last four to 18. This configuration
has a probability of occurrence [1] to 1/11.66.
Thus, 91.43% of possible combinations of onset did not this ratio.
Figure
3 analyses the constant 1/Pi (1/π). The
same phenomenon is observed for this constant. The probability [2] that such a phenomenon occur simultaneously
for a constant and its inverse is to 1/23.33.
Only the constant Phi (φ), by its arithmetical nature, has of course this property.
The same phenomenon (Fig. 4) of ratio to 3/2 (27/18) is present in the
constant Phi (φ) and of course in
1/φ.
Also, there is determined (Fig. 5) that the ten digits of constants 1/π and 1/φ split
identically in both fractions to ratio 3/2: the same first six and last four
digits.
This double shape has only one likelihood of appearance [3]
to 1/210. So, 99.52 % of combinations of appearance of figures do not have this
shape.
2.2. The ratio 3/2 inside
other constants.
This phenomenon of ratio to
3/2 (27/18) is present in other significant constants. This arithmetical phenomenon
is not therefore haphazard. This phenomenon is present in constants_{}, ζ (5) (Zeta 5 function), number e (constant of Neper),
in
constants of Copeland and Kaprekar. Also, in significant fractions relating
directly to the decimal system as the fraction 9876543210/0123456789.
One note that, as for constants 1/π and 1/φ, the ten digits of the constants grouped in Figure 8 are
distributed identically in the two fractions of the ratio 3/2 with the same
first six and last four digits, although there are 210 possibilities [3] for the division into six and four figures in order of
appearance of digits in their decimals suite.
It will be shown later (Chapter 5.3) that this
combination of six and four digits is not random but occurs by much greater
propensity than is possible in according to probabilities.
3.
Areas by 1, 2, 3 and 4 figures in the fundamental constants.
π,
1/π, 1/φ and other constants (see 3.1) share
another peculiar arithmetical property. Alongside the
phenomenon of ratio to 3/2, their digits are divided to form four areas of
occurrence which are always by sums of multiples of number 9:
In these constants, the sums
of digits of four areas of
appearance (which size is regularly progressive) are always by multiples of the
number 9. These zones are formed by 1, 2, 3 and 4 ranks of digits appearance. Also,
these areas (see fig. 10) are always identical in according to the occurrence
rank:
- area by 1 figure: rank 4
- area by 2 figures: ranks 2 - 3
- area by 3 figures: ranks 1 - 5 - 6
- area by 4 figures: ranks 7 - 8 - 9 -10
This number 9 is the greatest divisor of 45, the sum of the ten digits of the decimal
system. The likelihood of appearance of this arithmetical arrangement [4] is only to
1/420 for every constant. 99.76 % of possible combinations do not have this
configuration. It seems therefore not very unlikely that precisely, Pi, Phi and their inverses
share this property.
3.1. Other constants with the
same properties.
Inside
constants which are presented figure 11, We note that, always with the same
probability of 1/420 and as for π, 1/π, 1/φ, the ten digits are by the same four arithmetical areas so as to form
four values which are multiple of 9. This with a ratio to 3/2 between the first
six and last four digits which occurred:
3.2. Similarity between the
constants 1/π and 1/φ.
About
constants 1/π, and 1/φ, it has been demonstrated that, in order of first
appearance places of digits of their decimals, both have the same ratio to 3/2,
also, both in this division the same first six and last four digits, both
spread their digits to form the same four arithmetical areas which are multiple
of 9. It is finally that, for these two fundamental constants, the same figures
appear in the same four areas of 1, 2, 3 and 4 digits. The probability [5]
of the occurrence of such a arithmetical phenomenon is only to 1/12600.
So,
the two most prime mathematical constants such as Pi and the Golden Number are
they bound by these strange phenomena. The order of their decimal has nothing
random about all that arithmetical phenomena similar to recur in other
significant constants. The same
phenomenon occurs (probability to 1/12600) between the constant ζ(5) (Zeta 5
function) and the number which is the decimal complementarity of _{}.
4. Similar phenomena
with other constants.
4.1.
Constants , and _{}.
A similar phenomenon appears
for constants , and _{} which are three fundamental constants of
mathematics: the square roots of the first three prime numbers. As in p, in numbers , and _{},
sums of the same groups described above (4 areas of digits occurrence) have always
values which are by multiples of the same number: 3 to , 5 to and 9 to _{}.
These three values are the three possible divisors of 45, which is the sum of
ten figures of decimal system. The probability of occurrence [6] such configurations
is to 1/18 and only 5.55 % of all possible
combinations (digits occurrences) have these properties.
It is remarkable that this phenomenon occurs precisely
for Pi, Phi (their inverses also) and for the square roots of the first three
primes (prime number after having this feature is the 103 number located in
27th position in the sequence of primes).
It also notes the increasing
order of the divisor for these three constants: 3 to , 5 to et 9 to _{}.
4.2. Variants
of constants and .
Two
variants of the constants and
are
organized in remarkably identical configurations. Their four arithmetical areas
(identical
to those defined above) are multiples of the same
divisor (3) and their main ratio (6/4 digits) is the same (11/4).
4.3.
Constant
The sum of the ten digits of
the decimal system is 45, the average of these ten figures is therefore 4.5
Remarkable phenomena appear in the constant .
This
constant has the same general phenomena described in this paper: prime ratio
whose two quotients (here 15/30) are multiples of a divisor of 45 and the same
groups of 1, 2, 3 and 4 figures which are by multiples to the same divisor of
45 ( here 5). But also, two other strange phenomena emerge.
First
phenomenon: the first six digits (0 to 5) of the decimal system is precisely
the group of top six. The probability [3] of occurrence of this combination is to 1/210. Second
phenomenon: from the first to the tenth place, the figures are so perfectly
symmetrical, forming groups of two numbers whose total is always equal to 9.
The arithmetical probability [7] by which this occurs
is to 1/945.
4.3.1.
Constants and ((π − 2)/π)^{2}
In the development of its
decimal digits, the constant has a similar arrangement to the number derived from
Pi, ((π − 2)/π)^{2}. This number is the result of
equation:
This
number is not arbitrary, this equation is similar to the equation (where x = 5):
By a probability [5] to 1/12600, these two numbers are organized with the
same digits in the four defined areas appearance:
4.4.
Other notable constants.
In
the constant the first six digits (0 to 5)
of the decimal system are divided into the top six of onset. This phenomenon
has a probability to occur than 1/210. However it is observed the same
phenomenon in the other five constants, variations of π, described Figures 18 and 19. Also,
as in , these numbers have the same configuration property into
four arithmetical areas which are by multiples to a divisor of 45. The
probability of such an arrangement is to 1/1050 [9] for each number. Also, with
a probability [5] to 1/12600, constants _{}_{ }and 11/π^{2 }have (as 1/π and 1/φ) the same distribution of figures inside the four defined arithmetical
areas.
Together constants
and ((π − 2)/π)^{2}, these five other constants, variants derived from π, φ and e, therefore have the same first six and last four
digits. The number
0.0123456789101112... , which is the concatenation of the sequence of integers,
has obviously his first six digits identical to those numbers. In these, the
appearance of ten digits of decimal system are organized also themselves into
four arithmetical areas previously defined:
This is surely no accident and
must be connected with all phenomena introduced in this article. Thus, the number
0.01235711131719
, concatenation of the sequence of prime numbers, with more
numbers 0 and 1, also organized themselves in four same arithmetical areas of a
multiple to divisor of 45 (also 3):
* Number is the hypotenuse of
a triangle whose sides are p and e:
Also, the sine value of this
angle (tan = e/π) has remarkable properties:
In
this sine, the digits apparitions are configured with the same four areas of a multiple
to divisor of 45 also (here it is 3). The first six and last four digits are
the same as in the constant (probability
[3] to 1/210). One can also note the unusual regular order
of digits occurrence: from 6 to 0 and
from 7 to 9.
5.
Other constants.
5.1.
Constants by ratio to 3/2.
Respective decimal
complementarities of π, 1/π, φ and 1/φ are: 4 − π, 1 − (1/π), 2 − φ and 1 − (1/φ). It is (arithmetically)
usual in these additional numbers, that digits look into the same
configurations described above by four areas which are by multiples to 9 in a
ratio to 3/2. However, it is quite strange that the variations of these numbers
presented as Figure 25 all have a ratio to 3/2 in order of first appearance of
their digits:
Variants _{} and _{} (identical variants of decimal
complementarities of φ and π) divide
respectively their six first and last four digits as in decimals of 1/π and of _{} (constant whose Phi is derived): probability [3]
to 1/210. These two combinations of six and four digits (see below in 5.3) are
distinguished by their propensity to appearances in all phenomena presented in
this article. Thus, two other formulas, trigonometric configurations and
identical variants of Pi and Phi, introduce a remarkable phenomenon. By a ratio to 3/2, digits occurrences of the square of
the sine of the angle whose tangent equals to π and those of the square of the sine of the
angle whose tangent equals to φ respectively
also fall with the same first six and last four digits that decimals of 1/π and of _{} (constant whose Phi is derived): probability [3]
to 1/210.
5.2.
Constants by four areas which are in multiples to 9.
By a
prime ratio (6 and 4 classed digits) to 3/2, in constants, which are variants of π, φ and e, introduced Figures 27, occurrence order of digits organises into the same four arithmetical areas
which are by multiples to 9 as those of π and
φ (probability [4] to 1/420).
The two first
variants in Figure 27 have same first six and last four digits that decimals of
1/π and of 1/φ : probability [3]
to 1/210. Third presented variant has same distribution of six and four digits
as constants _{}, ζ (5), etc. (probability [3] to 1/210 also). These two distributions of figures are unusually more frequent in the constants
presented here:
0 1 3 6 8
9 / 2 4 5 7 (1/π, 1/φ, etc.)
and
0 2 3 6 7
9 / 1 4 5 8 (_{}, ζ (5), etc.)
In Figure
27, the first two constants have in common to have the same distributions of
digits occurrence in their four defined arithmetical areas. This distribution
is identical to the value 1 (1/π) which is the decimal complement of 1/π. With a probability of respectively occurrence [5]
to 1/12600, these three numbers are organized with the same digits in the four
defined appearance areas. Of course, decimal complementarity of Phi has the same property (see 3.2):
The
values 1/(4φ) and log7 distribute identically also their 10 digits in these same four
occurrence areas (see Figure 27).
Also,
always with the same very low probability [5] to 1/12600, the last two trigonometric values of Figure 27 have the
same common feature:
5.3. Two preferred combinations.
In
connection with the phenomena presented in 4.5.1, both trigonometric configurations
of Figure 30 (* and ** in Figure 27), variants of 1/π, are a common
phenomenon also. The digits occurrences of the inverse cosine of the angle whose tangent
is equal to 4/π and those of the inverse cosine of the angle whose
tangent is equal to e/π fall respectively with the same first six and four
last digits as in decimals of 1/π and of _{} (constant used to form φ): probability to
1/210 [3].
The
likelihood of a combination of six and four digits is therefore only to 1/210,
so that 99.52% of the combinations of figures occurrence are not the same
configuration (first 6 and last 4 figures). However
it appears in the phenomena presented in this article, only two combinations of
appearances of digits in the constants are much more frequent than is possible
by these arithmetical probabilities. These two combinations of six and four
digits are (digits ranked in ascending order):
There
exists a singular relationship between Pi and Phi for the emergence of these
two preferred combinations. Indeed, two pairs of two identical formulas using
respectively π and
φ have their
digits occurrence which included in
these two combinations (formulas presented above in 5.1):
These
two combinations of six and four figures occur therefore into variants of Pi
and of Phi but without automatic respectivity for Pi or for Phi. Indeed, as
described in Figure 32, these two combinations are interchangeable in relation
to Pi and Phi. For example, three other formulas in connection with
trigonometry and linked either to Pi or to _{} produce
numbers with one or other of the two preferred combinations of digits
occurrence:
Each
of these two combinations has a probability of occurrence to 1/210, however
many constants presented here and not all related to Pi or to Phi part of one
or other of these basic combinations (023679/1458 and 013689/2457). Also, many are by arrangement in four arithmetical
areas of multiples to 9 and a prime ratio (six and four classified digits) to
3/2. Among 3 628 800 possible combinations, only 1 152 combine these criteria
for the one
or the other basic
combination. This is a probability to 1/3150. Figure 34 lists the constants
presented in this paper and who possess these properties.
5.4.
Attempting to explain the phenomena.
Study of number x, which is the result [15] of the equation*:
x^{3} 2x = (2φ 1)^{2}_{}
* which can also be written:
x^{3} 2x 5 = 0
It
will not show in this article why the phenomena presented. The author does not
a arithmetic explanation which is quite clean. However, research tracks can be
envisaged. For example, many of these arrangements appear in singular
trigonometric and/or geometric configurations. The author tries to close the
phenomena also by links either with the configuration of digits occurrence (for
example: same first 6 and last 4 digits) or with the nature of constants or
also with these two parameters.
Here
it is an example of a research approach that gives other peculiar results. Number x, which is the result [15] of the equation x^{3} 2x =
(2φ
1)^{2 } (shown above in Figure 7) produces,
by a derived formula, a number which has the same organization of first
appearance of figures as into Pi. These
two numbers have the same digits in the four areas of occurrence. Recall that
the probability [5] of such a
phenomenon is only to 1/12600 and so 99.99% possible combinations have not this
configuration. This number is 4/(x − 1):
This
result is similar to that presented below in 6.1 where the number 4/(e − 1) (recall e = Neper constant) shares
the same phenomenon along with the not fortuitous fraction 9876543210/0123456789. Also, the number _{} shares the same
phenomenon along with number (the inverse cosine of the angle whose tangent
is e/π). Both numbers were
in fact the same digits in their four respective occurrence areas and also
their first six and last four digits are those (like the number 4/(e − 1)) of one of the
preferred combinations described in the previous chapter:
Also, this number x, which is the result [15] of the equation x^{3} 2x =
(2φ
1)^{2}, generates other strange phenomena. It may be noted that both
values in Figure 37 (shown above in Figure 6) the same first six and last four
digits appear:
Value _{} is a geometric value : the perimeter of
the square with surface which is equal to Pi. The second value is algebraic [15] and is the result of equation x^{3} 2x (2φ 1)^{2} =
0 (or
x^{3} 2x 5 = 0). Substituting, in this equation,
x by _{ }and then to a second value x par _{} there is obtained two other numbers, too, with in order of appearance of the
digits of their decimal, a ratio to 3/2:
Also,
the respective order of occurrence for these two new numbers is not random. The
first value is organized with the same first six and last four digits as into
constants 1/π and 1/φ. These figures are organized into four areas defined
above by multiples of a divisor of 45 (here 3). For the second value, the first six
and last four digits are identical to the two original values (Figure 37).
Also, there are those first six and last four digits in the numbers (described
above) (2 φ)^{2} and 1/4φ. The author does
not explain these phenomena but believes they cannot be timely and that this is a research way.
5.5.
Other constants by four areas which are multiples to divisor of 45.
5.5.1.
Variants of Phi.
In variations of Phi, whose
three geometric values, the digits occurrence also organized into four
arithmetical areas (defined above) which are by a multiple to divisor of 45:
5.5.2.
Other constants.
Other mathematical constants also organize the first appearance of their
digits into the same four areas (described above) by a multiple to
a divisor of 45:
Also more, the fractal
dimension of the Cantor set (log2/log3) and two variants of it, a variant of Pi
and the fraction 631764/13467:
631,764
is a Kaprekar number and the number 13,467 is the number obtained by classing
the digits which compose it (digits taken once). Also, the number 631764/13467
is organized with a ratio to 3/2 (27/18), as the number 1467/6174, another
fraction incorporating a number of Kaprekar described above in Figure 6.
Also,
remarkable variations of the number 8 (2^{3}) still
organized into the same four areas:
6.
Variants of e (the Neper constant).
6.1.
Variants of e.
In two variants of e described above, _{}(reverse sine of
angle whose tangent is e/π) and (sine of angle whose tangent is 4/e), first digits
occurrence organized into four areas by multiples of 9 and in a ratio to 3/2.
The first variant has the same
first six and last four digits as one of the preferred combinations described
in Chapter 5. Variant 4/(e − 1) has exactly the
same properties:
Also,
with a probability to only 1/12600, variant 4/(e − 1) distributes the same digits in the four defined areas
as the not fortuitous fraction 9876543210/0123456789 :
Also, variants 4/(e + 1) and 1/(e − 1) are organized into four
arithmetical areas by multiples to a divisor of 45. The variant 1/(e−1) is organized into areas by
multiples to 9 whose the probability [11] to occur is only to 1/350:
The constant e has thus three
variants whose first digits occurrence of their decimals is organized into the
four described previously arithmetical areas by multiples to 9 (in ratio to
3/2):
It seems unlikely since the
views of many other phenomena in this article that these three arrangements are
by chance.
6.2.
Variant integrating Pi, Phi, e and i.
It was shown earlier that in
many variants of Pi, Phi and e, the first digits appearance of the decimals is organized
into two arithmetical preferred areas (see 5.3) in a ratio to 3/2. A formula
incorporating these three constants produces a number whose the first digits
appearance of the decimals is organized in two arithmetical areas with
precisely one of these two combinations of numbers. In this constant, the first
six and last four digits are identical to constants 1/π and 1/φ.
Besides this formula incorporating
Pi, Phi, e, but also the imaginary number i, four
fundamental mathematical constants, generates a number whose the first digits
appearance of the decimals is organised into the same four arithmetical areas
by multiples to a divisor of 45 as those described
above. This is the formula, variation of a continued fraction of
Rogers-Ramanujan:
For to simplicity the next demonstrations, there will name r (r as Rogers and Ramanujan) this formula
incorporating four fundamental mathematical constants.
Many numbers derived from this formula produce phenomena similar to
those described throughout this article. So number _{} has the same arithmetical arrangement as r and its first six and
last four digits are identical to r (and to 1/π ,1/φ, etc.). Still with a ratio to 3/2 and a
arrangement into four arithmetical areas which are previously defined, number _{} described similar arrangements:
Also,
with a probability to only 1/12600, r
and variant 1 (_{}) (which is
decimal complementarity of_{ }_{}) distribute their own digits in the four defined areas.
The six numbers _{}, _{}, _{}, _{}, _{} and _{}, all
variants derived to r,
have the first digits occurrence of decimals organized into four
arithmetical areas by multiples to a divisor of 45:
Recall
that only one combination of digits occurrence onto eighteen has this property
and that 94.44% of possible combinations have not this configuration. Also, the
last two numbers presented in Figure 49 are organized, with a probability [3] to 1/210, with the same first six and last four digits.
7. Phi+ : a number which is a
cousin of Phi.
A number which is a variant of
the Golden Number (Phi) has some remarkable properties which are directly in
connection to the phenomena introduced above. The Golden Number is given by the
formula:
Substituting, in this formula,
the square root of 5 by the cube root of 5 there is obtained the number:
= 1,3549879733383484946765544362719
0
This number has the same
arithmetical arrangements as Phi: in these, the appearances of the figures are
organized in the same four occurrence areas (described above) to form sums
whose values are by multiples of 9. The probability [11] that the digits
occurrences are organized into this four areas of multiples of 9 is only to
1/350. 99.71% of all possible combinations of the appearance of ten digits of
the decimal system have not this arithmetical arrangement. There is very
unusual and not fortuitous that Phi [(_{}+1)/2] and Phi+ *
[(_{}+1)/2] possess these properties simultaneously.
*This number is provisionally
named [12] Phi+ and it is written φ_{+}. This number creates many other remarkable numbers in many derived forms.
Variants of this number presented below have the same singular arrangements as those
described above in the article. Many of these variants have unusual connections
with Pi and Phi (the Golden Number).
7.1. Formula 2(φ_{+}^{2 }+ φ_{+})
So,
formula 2(φ_{+}^{2 }+ φ_{+}) gives the number :
6.3819607624598270114596114251567
This
number has the same arrangement of digits occurrence into four areas by
multiples of 9 and a ratio to 3/2 as the constants 1/π and 1/φ (probability [4] to
1/420). Too, in this distribution, its first six et last four digits are the
same as in 1/π and 1/φ (probability [3] to 1/210).
Still
more, with probability [5] to 1/12600, This number organized with the same digits occurrence into
the four appearance areas as numbers 1-(1/π), (π_{})^{4 }and
cosine reverse of the angle whose tangent is equal to 4/π (three numbers
described above in 5.2).
7.2. Formula 8 2(φ_{+}^{2 }+ φ_{+})
By subtracting the number 2(φ_{+}^{2 }+ φ_{+}) from the superior second whole number (so from
8) there is obtained a number very
similar to the Golden Number * (but not the Golden Number):
8 2(φ_{+}^{2 }+ φ_{+}) = 1.6180392375401729885403885748433
*φ = 1.61803398874989484820458683436564
This number is organized in its digits appearances, just as the Golden
Number:
Thus, a variation of Phi+ [12] produced a number almost identical to
Phi, whose appearance of its decimal digits is almost identical to Phi, but
that is different from Phi. This reinforces the idea that the organization of
digits occurrences in the fundamental constants is not by chance.
7.3. Formula 1 (1/_{})
The formula 1 (1/_{}) gives a number whose the organization of
digits occurrences is very close to Pi:
= 0,4151964523574267868
0
This number has the same appearance of the first six and last four
digits as Pi. Also, it is organized as Pi in four areas by multiples of 3:
7.4. Formula 1 (_{}/2).
The formula (_{}/2), close formula
to Phi+, gives the number:
= 1.8549879733383484946765544362719
0
This number has the same arrangement of digits occurrence into four
areas by multiples of 9 as Phi+.
The formula 1 (_{}/2), which is the
decimal complementarity of the previous, gives a number whose the organization
of digits occurrences is very close to Pi:
Still
more, with probability [5]
to 1/12600, This number 1
(_{}/2) is organized with exactly the same four digits
areas as the number 3/[(4/p)^{2} +1], a variant of Pi whose the arrangement of digits occurrences
is very close to Pi also :
This number 3/[(4/π)^{2} +1] is not fortuitous. A number whose the formula is very
close has its digits occurrences organized into a oddly close configuration.
This is the number 4/[(4/π)^{2} +1]:
7.5.
Other derived formulas to Phi+.
Formulas φ_{+}^{2 } φ_{+} and 1/(φ_{+}^{2 } φ_{+}) are organized, in the occurrence of its
digits, into four areas by multiples to a divisor of 45 (here 3). This is with
a probability [6]
to 1/18.
With the same occurrence
probability [6]
to 1/18, formulas presented Figure 58, variants of Phi+, are organized in the
same configurations into four areas by
multiples to a divisor of 45:
The last formula (φ_{+ } 1)^{5}/φ_{+}^{5}, ratio of the
first two, has the same first six and last four digits as the constant _{} and other numbers presented in 4.4 which, remember
this, split their first six digits from 0 to 5 and their last four from 6 to 9.
One can also note the unusual regular order of digits occurrence for this
formula: 0-1-2-3-4-5 and 9-8-7-6. What makes that this number has the same
digits in the four defined arithmetical areas (probability [5] to 1/12600) as
the concatenation (presented in 4.4) of the integers sequence (0,01234567891011
).
The formula φ_{+}^{2}_{ }/(φ_{+}^{2}_{ } 1), with a probability [3] to 1/210, has the same occurrences of first six and last four digits as
numerous constants introduced above in this paper which mainly 1/Pi, Phi, etc. This formula can be
closed to the trigonometric formula π^{2}/( π^{2} + 1^{2}) presented
above in 5.1. For indeed, this formula can be written with the imaginary number
i :
7.6.
Phi+ and The hard hexagon constant.
The
number (_{}+1)/2 (which is
Phi+) and the number _{} are organized with the same first six and last
four digits. Still, this combination of six and four digits is the same as in The hard hexagon constant [14]:
Note:
into the square root of ''The hard hexagon constant'' [14] (1.18130689174291315
), appear
the same first six and last four digits as into the numbers 1/π and φ (one
of two preferred combinations of occurrences described above in 5.3). This is yet in all likelihood not a fortuitous phenomenon.
7.7.
Phi+, Phi and e.
Some
variants of Phi+ associate to Phi show other strange phenomena including
unusual similarities to variants of the constant e (Neper constant):
8.
Other findings.
In order not to overload this article
by too of many shows, the author has here presented the findings only which
have most significant connections with the phenomena described. Here are just
some examples of investigations reinforcing the idea that the first appearance
of the ten decimal digits inside remarkable constants is not random.
8.1.
Landau-Ramanujan Constant.
The Landau-Ramanujan Constant
itself organizes by a ratio [1] to 3/2 (27/18). Three variants of
this constant (named C here) have the same property whose constant C^{2} which
itself organizes besides into four
arithmetical areas by multiples of a divisor of 45:
8.2.
Number 33 and number Pi.
The order of first appearance
of the ten digits of the square root of 33 generates four arithmetical areas
previously defined (by multiples of a divisor of 45). In association to Pi,
this number gives some others with similar characteristics:
* The last formula in Figure 63
is not a variant to 33 but its arithmetic construction _{}is close to formula
_{}.
Still, inside this number, the first six and last four digits occurrences are
the same as one of the two preferred combinations previously highlighted in
5.3.
8.3.
Ratio 1/7.
Many
rational numbers are formed by a sequence of repeated decimals, this is one of
their characteristics. Very often this repetitive sequence consists to digits
whose the sum is a multiple of 9. The first rational number (among inverses of integers) to be
formed from such a sequence is the number 1/7 whose the repeated sequence of
its decimals is formed by the digits 1-4-2-8-5-7 (1/7 = 0.142857142857
). The
addition of these six different figures gives 27 and adding the four missing
numbers (0-3-6-9) gives 18. This
gives a ratio to 3/2 between these two sets of digits. In ranking order of
magnitude the first six digits gives the number 124,578 and the ratio
124578/142857 gives a number whose digits are formed by repetitive series
8-7-2-0-4-6. This sequence is organized into three areas by multiples of 9
which are identical
to those described in this paper and the four missing digits form a fourth area
by a multiple of 9 and in a prime ratio to 3/2 with this series:
The author find that this phenomenon
is not accidental and is linked to all other phenomena introduced in this
article. So, this is a possible research way to explain these singular
phenomena.
8.4.
The Fibonacci series.
By
dividing each number in the Fibonacci sequence by 10^{n}, where n is
the rank of each number, then summing these numbers, there is obtained a number
that tends toward the rational number 10/89. This number has the same
organization of digits occurrence into four areas by multiples to a divisor of
45 also:
It
is well known that the Fibonacci sequence gives the number Phi, main topic of
this article. This further demonstration gives credence to the idea that the
order of first appearance of the ten digits forming the decimals of numerous
mathematical constants is not fortuitous.
9.
Prime numbers, decimal system and 3/2 ratio.
In parallel to the study of
the order of the first occurrence of the ten digits in decimals of many
mathematical constants and particular numbers described in this paper,
remarkable properties about the formation of the ten digits and the scripture
of primes numbers are obliged to be introduced here.
9.1.
Formation of the ten digits (according to prime numbers).
The ten digits: 0 1 2 3 4 5 6
7 8 9
Six primes or
fundamental numbers (0 and 1): 0 1 2 3 5 7. Sum equal to 18
Four not primes and
not fundamental numbers : 4 6 8 9. Sum equal to 27
So a ratio to 18/27 (so 2/3).
Four not primes: 4 6 8 9
The six primes or fundamental
numbers are (of cause) just combinations of 6 primes (themselves).
The four not primes
are combinations of 9 primes (see fig. 66).
So a ratio to 6/9
(so 2/3).
Sum of the primes
to form the six primes or fundamental numbers
= 18
Sum of the primes
(one time enumerated) to form the four not primes = 12
So a ratio to 18/12
(so 3/2).
9.2.
Digit scripture of the primes
All prime numbers have only
digits 1 2 3 5 7 - 9 in latest position so 6 digits are possible.
All prime numbers have not
digits 0 4 6 8 in latest position
so 4 digits are not possible.
Sum of 1 2 3 5 7 9
is equal to 27 and sum of 0 4 6 8
is equal to 18.
So there are two ratio 3/2
about digit scripture of the primes : 6 and 4 digits possible or not possible
in latest position and 27 and 18 the sums of these 6 and 4 digits.
10.
Conclusion.
The order of the first
occurrence of the ten digits forming the decimals of many mathematical
constants is not random. Into the constants which are introduced in this paper,
always identical areas of one, two, three and four digits have sums which are
by multiples to a same divisor of 45 (according to the constants: 3, 5 or 9).
This occurrence areas are always : in the occurrence rank 4 for the one
digit area, in the occurrence ranks 2 and 3 for the two digits area, in the
occurrences ranks 1, 5 and 6 for the three digits area and in the occurrence
ranks 7, 8, 9 and 10 for the four digits area:
The
occurrence probability of this basic configuration is only to 1/18 and 94.44 % of possible configurations have not this arrangement.
However, the constants Pi, 1/Pi, Phi (and 1/Phi), numbers , and _{},
number (square root of the average of ten digits of the
decimal system), The Zeta 5 function and very many variants of these numbers
here introduced whose Phi+ [12] and some variants of the Neper constant (e) are organized
into this basic configuration. A large proportion of these numbers are values
related to the geometry field.
Also, a high proportion
(higher probabilities) of these numbers, including the major Neper constant (e) has a ratio to
3/2 in the digits appearance of their decimals (six first against four occurred
digits).
The number Pi and
the Golden Number (Phi) possess these properties and they have particularity to
reproduce these arithmetical faculties for their inverses. The inverse to
number Pi and the inverse to Golden Number are closed by a more still singular
phenomena because, for these two fundamental constants of Mathematics, by a
probability to only 1/12600, the same
figures occurs into the four defined digits occurrence areas of their decimals.
Also, the observation that these singular phenomena
are verified for many other constants, whose the numbers , and _{} (square roots of the first three prime
numbers) and for variants of the Neper constant (e) confirms that the order of
first appearance of digits in the decimal of constants which are presented in
this paper is not random.
In conclusion, the author proposes to consider the existence of a new
family of numbers having the characteristics described in this article. Family
of numbers which the number Pi and the Golden Mean are the most significant
representatives. Also, the author recalls and insists that this new field of
research investigates, in numbers, only the first appearance of the ten digits
of the decimal system and suggests that it is not fruitful to extend the
investigations to the following appearances. The fact, not presented here but
experienced by the author, that these investigations are sterile paradoxically
reinforces the idea that the phenomena introduced in this study must be subject
to greater attention.
Since the publication of this article, the author continues his
researches on these intriguing arithmetic phenomena about the first occurrences
of the ten digits of the decimal system in other significant numbers and
constants. These new investigations are presented here:
It
is in an unstructured form. These new investigations will be gradually integrated
into a future version of the article. The author invites anyone to participate
in these new researches. If you think you've discovered some interesting thing
in link to this, you can introduce your own discoveries to the author. They will then be
presented (with your references) on this site.
Annexe
[1] There are
3,628,800 different combinations in the distribution of digits occurrences in
decimals of constants. 311,040 combinations have a ratio to 3/2 (27/18).
This is only to 1/11.66 and therefore
91,43 % of the possible combinations are not this ratio.
[2] The
probability that the constant π and 1/π have simultaneously a ratio
to 3/2 (see [1]) is to 1/23.66.
[3] Among the
3,628,800 different combinations, 17,280 have the same distribution of 6 and 4
digits, this is only to 1/210 and therefore 99.52 % of digits occurrence
combinations have not the same configuration (of 6 and 4 digits).
[4] Among the
3,628,800 different combinations, 8,640 have the same arithmetical
configuration into 4 areas by multiples of 9 and a ratio to 3/2. This is only
to 1/420 and 99.76 % of possible combinations have not this configuration.
[5] Among the 3,628,800
combinations, only 288 have the same digits distributed into the 4 defined arithmetical areas. This is only to
1/12600 and 99.99 % of possible combinations have not this
configuration.
[6] Among the
3,628,800 combinations, 201,600 have the same 4 areas of digits whose the sums
are by multiples of the same numbers (3, 5 or 9 in according to combinations).
This is only to 1/18 and 94.44 % of possible combinations have not this
configuration.
[7] Possible
combinations = 9 x 7 x 5 x 3 x 1 = 945.
[8] - sin^{2}
of angle whose tangent = π : 0.908000331649624767544
π^{ 2}/( π^{2} + 1)
- sin^{2} of angle whose tangent
= φ : 0.72360679774997896964091
5
φ^{2}/( φ^{2} + 1)
[9] Among the
3,628,800 combinations, 3,456 have in same time the first 6 digits of decimal system
(from 0 to 5) in the first six ranks of occurrence and the same areas of four
digits whose the sums are by multiples of the same divisor of 45 (3 or 5 in
according to combinations). This is to 1/1050.
[10] - (π_{ }_{})^{4 }: 876.681819306021935127962994198
- 1/cos of angle whose tangent
is 4/π : 1.61899318660623286240765967
- 1/cos
of angle whose tangent is e/π : 1.32237207696748056509441395
- 1/4φ : 0.154508497187473712051146708
- 3φ/2 : 2.427050983124842272306880
- sin of angle whose tangent is 4/e : 0.827091663
70615584
- sin of angle whose tangent is _{}: 0.822701898389593218034076
[11] Among the
3,628,800 combinations, 10,368 have the same arithmetical configuration into 4 areas
by multiples of 9 (with or without ratio to 3/2). This is to 12/350 and 99.71 %
of possible combinations have not this configuration.
[12] Pending
to a more formal name, the author proposes to temporarily call the number (1.3549879733383484946765544362719
0
), variation of the
Golden Number, Phi + and to represent this one by the symbol φ_{+}.
[13] r = : for to simplicity the demonstrations, this formula
is named r as Rogers and Ramanujan.
_{} = 1.6210555640245749558387576785698
_{} = 1.6150180455567689912054319315883
_{} = 1.6240827818983623986718091080939
5
_{} =
0.14562608632223968713316326201939
_{} = 1.6195440717201534561999262513712
8
_{} = 1.6190405542023005829273871877919
[14] The hard hexagon constant : the
informed reader must know that constant whose author (self educated researcher)
found no clear definition.
[15] x is the result of the equation x^{3} 2x =
(2φ 1)^{2} which is equal to the
function x^{3} 2x 5 = 0 which is used in
the
Pi and Golden Number:
not random occurrences of the ten digits. Jean-Yves BOULAY 2008-2012©