It has been conjectured that the sequence @math modulo @math is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we describe an algorithm to compute @math modulo @math to @math . We then statistically analyze its distribution. Our results strongly agree with the hypothesis that @math modulo 1 is uniformly distributed.
To our knowledge no research has been published to evaluate numerically the distribution of the sequence @math modulo 1. However, the distribution of this sequence has been studied in relation to another topic: Mahler's Z-numbers. A Z-number is a real number @math such that @math for all @math . Mahler conjectured in 1968 that no Z-numbers exist @cite_10 .
@cite_10
2155254221
Let α be an arbitrary positive number. For every integer n ≦ 0 we can write where is the largest integer not greater than , i.e the integral part of , and r n is its fractional part and so satisfies the inequality