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On the decimal and octal digits of $1/p$

Kurt Girstmair

Abstract

Let $p$ be a prime $\equiv 3$ 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 $0,\ldots,9$ 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 $1/p$.

On the decimal and octal digits of $1/p$

Abstract

Let be a prime mod 4, , and suppose that 10 has the order mod p. Then has a decimal period of length . We express the frequency of each digit 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 and for the octal digits of .

Paper Structure

This paper contains 3 sections, 6 theorems, 34 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Digits and quadratic residues
  3. 3. Octal digits

Key Result

Theorem 1

In the above setting, where the numbers $\delta_k$ take the following values. If $p\equiv 3$ mod 8, then If $p\equiv 7$ mod 8, then

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Theorem 4