The golden number to 17 000 000 000 digits

Definition:

When the ratio between two numbers is equal to the ratio between the biggest and the sum of both numbers, that ratio is called the golden number. It's noted φ (phi)

It satisfies the equation (1+φ)/φ=φ, which solves in φ=(1+√5)/2.

Like square root of 5, φ is an irrational number.

It means that its decimal extension i sinfinite and that the digits don't repeat following a periodic scheme.

The 100 first digits of the decimal extension are:

1.6180339887 4989484820 4586834365 6381177203 0917980576
2862135448 6227052604 6281890244 9707207204 1893911374...

The 100 "last" digits between positions 16 999 999 901 and 17 000 000 000 are:

...7284422710 4431184343 6942323356 8165603643 7818286382
5414151339 5284187492 1138675755 6615977802 6510689641...

Conjectures:

There are two main conjectures about phi's digits:

- Is φ a normal number?

A normal number is a number in which the probability of finding each digit is 1/10, finding each 2 digit sequence is 1/100, finding each 3 digit sequence is 1/1000, etc.

See data collected on the 17000000000 first digits

- Is φ a universe number ?

A universe number is a number where you could find whatever digit sequence you want , provided that you look sufficiently far.

See data collected on the 17000000000 first digits

Algorithm used:

First calculate square root of 5/4 with Newton's algorithm. Then, add 1/2.

The square root of 5/4 can be computed this way:

Use iteration x_n <- x_n + x_n*(5-4x_n²)/10 beginning with an aproximated value of √(5/4)
(for example the number 1)

Complexity of the calculation:

To calculate n digits, as the accuracy doubles at each iteration, log₂(n) iterations are needed.
Each iteration is essentially made of two multiplications.
If only the necessary accuracy is used at each iteration, then the calculation can be reduced to a constant number of multiplications (less than 3).
The multiply algorithm used is Strassen one. Its complexity is O(n log n).

Calculation completed on 02/29/2008 after 30 hours.

Verification:

To prevent any design, programming, computation, transmitting or storage error to be introduced into the digits, the following verifying procedure was used:

- calculate 17000000000 digits of φ.
- store the result in a file (2 digits per byte, i.e. 8 Gib).
- calculate an MD5 footprint of the file.
- re-read the file.
- calculate (φ-1/2)². It should be 5/4.
- measure real error, which should be less that the expected accuracy.
- As long as the file verifies the stored MD5 footprint, one can be sure of the exactness of the digits.

Verificacion completed on 03/01/2008 after 15 hours.

Computing configuration:

Program written in C language.

CPU: Intel Pentium dual core 64 bits overclocked at 3 GHz
1Mib of L2 cache, 4Gib DDR2 memory,
450 Gb of hard disk space used.

GNU/Linux Ubuntu 7.10 64 bits system,
gcc 4.2.1 compiler

Previous calculations:

1998 : 10 000 000 digits, Simon Plouffe

2000 : 1 500 000 000 digits, Xavier Gourdon and Pascal Sebah

2002 : 3 141 000 000 digits, Xavier Gourdon and Pascal Sebah

10/17/2007 : 5 000 000 000 digits, Alexis Irlande former page

24/11/2007 : 10 000 000 000 digits, Alexis Irlande former page

?/?/2007 : 10 000 000 000 digits, S. Kondo and S. Pagliarulo (information welcome)

The author, Alexis IRLANDE

I've got a PhD in computer science, I am presently assistent professor at the math department of the Universidad Nacional de Colombia, in Bogotá.
I am teaching (among other things) algorithms, cryptography, numerical analysis, networks, computation theory, programmation in C, differential equations, logic, combinatorics, probability and linear algebra.

Contact me: airlande at unal.edu.co
You will find my personal cell phone number at position 9467366848 in the golden ratio's digits.