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/π2 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*:

 

x3 – 2x = (2φ – 1)2

* which can also be written: x3 – 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 x3 – 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 x3 – 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 x3 – 2x – (2φ –1)2 = 0 (or x3 – 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 (23still 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/(e1) 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 π2/( π2 + 12) presented above in 5.1. For indeed, this formula can be written with the imaginary number : 

 

 

 

 

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 C2 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 10n, 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:

new discoveries

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] - sin2 of angle whose tangent = π : 0.908000331649624767544… π 2/( π2 + 1)

      - sin2 of angle whose tangent = φ : 0.72360679774997896964091…5… φ2/( φ2 + 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  x3 – 2x = (2φ – 1)2 which is equal to the function  x3 – 2x – 5 = 0 which is used in the Newton method.

 

 

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 Pi and Golden Number: not random occurrences of the ten digits. Jean-Yves BOULAY 2008-2012©