TITLE:
Fibonacci Congruences and Applications
AUTHORS:
René Blacher
KEYWORDS:
Fibonacci Sequence, Linear Congruence, Random Numbers, Dependence, Correct Models
JOURNAL NAME:
Open Journal of Statistics,
Vol.1 No.2,
August
1,
2011
ABSTRACT: When we study a congruence T(x) ≡ ax modulo m as pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that the dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We will use this result to obtain truly random number sequences xn. For that purpose, we will use non-deterministic sequences yn. They are transformed using Fibonacci congruences and we will get by this way sequences xn. These sequences xn admit the IID model for correct model.