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Pi and 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.

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Pi, 1/Pi and Phi: same arithmetical arrangements?

In constants π, 1/π and φ (a), the occurrence order of ten digits of the decimal system (c) compared to the rank of appearance (b) is organized into identical arithmetical arrangements (d).

Inside constants Pi and Phi, the four appearance areas have the same digits of the decimal system. And all these areas add up to multiples of 9.Pi and Golden Number

Pi and Golden Number not random occurrences of the ten digits

Constants square roots of 2, 3 and 5.

A similar phenomenon appears for constants square roots of 2, 3 and 5 which are three fundamental constants of mathematics: the square roots of the first three prime numbers. As in π, 1/π and φ, in these three numbers, 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 square root of 2 , 5 to square root of 3 and 9 to square root of 5. These three values are the three possible divisors of 45, which is the sum of ten figures of decimal system. The probability of occurrence such configurations is to 1/18 and only 5.55 % of all possible combinations (digits occurrences) have these properties.

square root of 2, of 3 and of 5

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).