Fibonacci Congruences and Applications

René Blacher

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DOI: 10.4236/ojs.2011.12015                

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 x_n. For that purpose, we will use non-deterministic sequences y_n. They are transformed using Fibonacci congruences and we will get by this way sequences x_n. These sequences x_n admit the IID model for correct model.

Keywords

Fibonacci Sequence, Linear Congruence, Random Numbers, Dependence, Correct Models

Conflicts of Interest

The authors declare no conflicts of interest.

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