Abstract of the paper "Pi an Golden Number: not random occurrences of the ten digits".
Number Pi and the Golden Section as well as the inverse of these numbers are made up of a series of apparently random decimal places. This paper is on the occurrence order of the 10 digits of decimal system in these fundamental mathematic numbers. It is in fact that the ten digits of decimal system does not appear randomly in the sequence of Pi and in Golden Section. Also, same phenomena operate in many other constants of which the square roots of numbers 2, 3 and 5, the first three prime numbers.
1. Introduction.
The number Pi (p) 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 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 (p) and the digits sequence of Golden Number (φ). The same phenomenon is also observed for the inverse of these two numbers (1/p 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...
(excerpt from the paper)...In constants π, 1/π and φ (a), the occurrence order of ten digits of the decimal system © compared to the rank of appearance (b) is organized into identical arithmetical arrangements (d)
(excerpt from the paper)...Into the occurrence order of digits of their decimals, the constants 1/π and 1/φ have the same ratio to 3/2 (probability to 1/11. 66).
In this division, there are the same first six and last four digits (probability to 1/210).
Both split their figures to form the same four areas multiples of 9 (probability to 1/420).
It appears finally that, for these two fundamental constants, the same digits appear in the same four areas of 1, 2, 3 and 4 figures (probability to 1/12600)…
(excerpt from the paper)...A formula, derived from the continued fraction of Rogers-Ramanujan including the 4 fundamental numbers π, φ, e and i, isorganized into same four zones of 1, 2, 3 and 4 digits and with the same first 6 and last 4 digits as the constant 1/π, and the constant 1/φ.
Complet paper here (and in attached file): http://jean-yves.boulay.pagesperso-orange.fr/pi/index.htm
Pi and Golden Number not random occurrences of the ten digits.pdf
Others some examples of phenomena introducted in the paper.
Abstract of the paper "Pi an Golden Number: not random occurrences of the ten digits". Number Pi and the Golden Section as well as the inverse of these numbers are made up of a series of apparently random decimal places. This paper is on the occurrence order of the 10 digits of decimal system in these fundamental mathematic numbers. It is in fact that the ten digits of decimal system does not appear randomly in the sequence of Pi and in Golden Section. Also, same phenomena operate in many other constants of which the square roots of numbers 2, 3 and 5, the first three prime numbers. 1. Introduction. The number Pi (p) 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 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 (p) and the digits sequence of Golden Number (φ). The same phenomenon is also observed for the inverse of these two numbers (1/p 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... (excerpt from the paper)...In constants π, 1/π and φ (a), the occurrence order of ten digits of the decimal system © compared to the rank of appearance (b) is organized into identical arithmetical arrangements (d) (excerpt from the paper)...Into the occurrence order of digits of their decimals, the constants 1/π and 1/φ have the same ratio to 3/2 (probability to 1/11. 66). In this division, there are the same first six and last four digits (probability to 1/210). Both split their figures to form the same four areas multiples of 9 (probability to 1/420). It appears finally that, for these two fundamental constants, the same digits appear in the same four areas of 1, 2, 3 and 4 figures (probability to 1/12600)… (excerpt from the paper)...A formula, derived from the continued fraction of Rogers-Ramanujan including the 4 fundamental numbers π, φ, e and i, isorganized into same four zones of 1, 2, 3 and 4 digits and with the same first 6 and last 4 digits as the constant 1/π, and the constant 1/φ. Complet paper here (and in attached file): http://jean-yves.boulay.pagesperso-orange.fr/pi/index.htm Pi and Golden Number not random occurrences of the ten digits.pdf Others some examples of phenomena introducted in the paper.
