Let p be a prime <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\equiv 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≡</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> mod 4, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and suppose that 10 has the order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(p-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> mod p. Then 1/p has a decimal period of length <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(p-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We express the frequency of each digit <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0,\ldots ,9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation> 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

期刊论文
发表日期：

2026年4月24日

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Kurt Girstmair

摘要

Let p be a prime $\equiv 3$ $\equiv 3$ mod 4, $p>3$ $p > 3$ , and suppose that 10 has the order $(p-1)/2$ $(p - 1) / 2$ mod p. Then 1/p has a decimal period of length $(p-1)/2$ $(p - 1) / 2$ . We express the frequency of each digit $0,\ldots ,9$ $0, \dots, 9$ 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.