2018/08/24 by Matthew Brand, Brand, Matthew · 1 voice
Mathematics · #11A15 #11Y55 #60C05 #60G40 #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1808.07994
openalex publication_date 2018/08/24 · arxiv published 2018/08/24 · arxiv updated 2018/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
How many fair coin tosses to choose 1 of n options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We propose a bit-efficient scheme, prove optimality, derive the expected number of coin tosses e[n], characterize its fractal structure, and develop sharp upper and lower bounds, both discrete and continuous. A minor but noteworthy corollary, with real-world examples, is that any lottery or simulation with finite budget of random bits will have a predictable pattern of lucky and unlucky numbers.