On the decimal and octal digits of 1/p

2025/10/09 by Kurt Girstmair, Girstmair, Kurt
Computer Science · Mathematics · #Coding theory and cryptography #Computability, Logic, AI Algorithms #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2510.07873

Abstract

Let p be a prime ≡ 3 mod 4, p>3, and suppose that 10 has the order (p-1)/2 mod p. Then 1/p has a decimal period of length (p-1)/2. We express the frequency of each digit 0,…,9 in this period in terms of the class numbers of two imaginary quadratic number fields. We also exhibit certain analogues of this result, so for the case that 10 is a primitive root mod p and for the octal digits of 1/p.

Citations

• An analogue of Girstmair's formula in function fields
• Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Related

• On the sum of digits of 1/M in \mathbbF_q[x]
• Distribution of residues modulo p using the Dirichlet's class number formula
• Power residues, digit expansions and relative class numbers
• Algebraic functions and class number formulas
• An analogue of Girstmair's formula in function fields