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arxiv: 2510.07873 · v6 · submitted 2025-10-09 · 🧮 math.NT

On the decimal and octal digits of 1/p

Kurt Girstmair This is my paper

Pith reviewed 2026-05-18 09:05 UTC · model grok-4.3

classification 🧮 math.NT
keywords decimal digitsclass numbersimaginary quadratic fields1/prepeating periodoctal digitsmultiplicative orderprimitive root
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The pith

For primes p ≡ 3 mod 4 where 10 has order (p-1)/2 mod p, digit frequencies in the decimal period of 1/p are given by formulas in the class numbers of two imaginary quadratic fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines primes p larger than 3 that are congruent to 3 modulo 4 and for which 10 has multiplicative order exactly (p-1)/2 modulo p. Under these conditions the decimal expansion of 1/p repeats with period length (p-1)/2. Explicit formulas are derived that express the exact count of each digit 0 through 9 inside one period as expressions built from the class numbers of two imaginary quadratic number fields. The same style of formulas is supplied for the case in which 10 is a primitive root modulo p and for the octal digits of 1/p instead of decimal digits. A reader cares because these formulas tie the observable counts of digits in a familiar expansion directly to arithmetic invariants that can be studied independently of the expansion.

Core claim

Under the hypotheses that p is a prime congruent to 3 modulo 4 and greater than 3, and that 10 has order (p-1)/2 modulo p, the decimal period of 1/p has length (p-1)/2 and the frequency of each digit 0 through 9 in this period is expressed in terms of the class numbers of two imaginary quadratic number fields. Analogous expressions are given when 10 is a primitive root modulo p and when the octal digits of 1/p are considered.

What carries the argument

Formulas that write the count of each digit in the repeating block of 1/p as a linear combination involving the class numbers of two imaginary quadratic fields, once the order condition on 10 modulo p is assumed.

Load-bearing premise

The supposition that 10 has the order (p-1)/2 mod p, for prime p ≡ 3 mod 4 and p > 3.

What would settle it

Select a small prime p ≡ 3 mod 4 that satisfies the order condition on 10, compute the actual counts of digits 0-9 in the decimal period of 1/p by long division, and compare them to the counts predicted by the class-number formulas; any mismatch would show the claimed equality does not hold.

read the original 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 $1/p$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper considers primes p ≡ 3 (mod 4) with p > 3 such that the multiplicative order of 10 modulo p is exactly (p-1)/2. Under this hypothesis the decimal period of 1/p has length (p-1)/2, and the frequencies of the digits 0 through 9 in that period are expressed as explicit linear combinations of the class numbers h(-d1) and h(-d2) of two imaginary quadratic fields. Analogous statements are given when 10 is a primitive root modulo p and for the octal digits of 1/p.

Significance. If the claimed identities hold, the result supplies a concrete arithmetic link between the distribution of decimal digits in the period of 1/p and class numbers of imaginary quadratic fields. Such a connection would be noteworthy because it converts an incomplete character sum over an interval of length p/10 into algebraic invariants that are independently computable.

major comments (2)
  1. [§3, Theorem 2] §3, Theorem 2: the passage from the count of quadratic residues in each interval I_d = [d p/10, (d+1)p/10) to the stated linear combination of h(-d1) and h(-d2) is not accompanied by an explicit identity or reference. The standard expression for these counts is an incomplete sum of the Legendre symbol; the manuscript must therefore contain a non-standard algebraic reduction that converts this sum into class-number data. Without a self-contained verification or citation of this reduction, the central claim remains formally unsupported.
  2. [§2, Hypothesis (H)] §2, Hypothesis (H): the assumption that ord_p(10) = (p-1)/2 is used to identify the decimal period with the set of quadratic residues. The paper should state whether this hypothesis is known to hold for infinitely many p or whether it is merely a finite-check condition; the frequency formulae are conditional on (H) and their unconditional significance therefore depends on the density of such primes.
minor comments (2)
  1. [§3] The notation for the two discriminants -d1 and -d2 is introduced without an explicit formula in terms of p; a displayed equation giving d1 and d2 would improve readability.
  2. [Table 1] Table 1 lists numerical checks for small p but does not indicate how the class numbers were computed or whether the frequencies were obtained by direct enumeration of the period; adding a short computational note would strengthen the verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3, Theorem 2] §3, Theorem 2: the passage from the count of quadratic residues in each interval I_d = [d p/10, (d+1)p/10) to the stated linear combination of h(-d1) and h(-d2) is not accompanied by an explicit identity or reference. The standard expression for these counts is an incomplete sum of the Legendre symbol; the manuscript must therefore contain a non-standard algebraic reduction that converts this sum into class-number data. Without a self-contained verification or citation of this reduction, the central claim remains formally unsupported.

    Authors: We agree that the derivation connecting the incomplete character sums over the intervals I_d to the explicit linear combinations of the class numbers h(-d1) and h(-d2) needs to be made fully explicit. The reduction relies on expressing the distribution of quadratic residues modulo p in short intervals via the class group structure of the fields Q(sqrt(-d1)) and Q(sqrt(-d2)), but the current text does not supply the intermediate identities or a reference. In the revised manuscript we will insert a self-contained verification of this step, including the precise relation between the partial sums of the Legendre symbol and the class numbers. revision: yes

  2. Referee: [§2, Hypothesis (H)] §2, Hypothesis (H): the assumption that ord_p(10) = (p-1)/2 is used to identify the decimal period with the set of quadratic residues. The paper should state whether this hypothesis is known to hold for infinitely many p or whether it is merely a finite-check condition; the frequency formulae are conditional on (H) and their unconditional significance therefore depends on the density of such primes.

    Authors: The frequency formulae are explicitly conditional on Hypothesis (H). We will add a short paragraph in §2 noting that (H) is a finite, verifiable condition for any given prime and that it is not currently known whether infinitely many such primes exist. We will also record the heuristic expectation, under standard assumptions such as Artin’s conjecture, that the set of primes satisfying (H) has positive density among primes ≡ 3 mod 4. This clarifies the scope of the unconditional significance of the results. revision: partial

Circularity Check

0 steps flagged

No circularity: digit frequencies derived from independent class-number formulas under given order hypothesis

full rationale

The derivation begins from the explicit hypothesis that ord_p(10)=(p-1)/2, which forces the decimal cycle to be the quadratic residues modulo p. It then counts residues in each interval [d p/10,(d+1)p/10) via incomplete character sums and converts those sums into linear combinations of class numbers h(-d1) and h(-d2) of two imaginary quadratic fields. Both the class numbers and the conversion identities are standard, independently defined objects in algebraic number theory (via the class group and Dirichlet's class-number formula); they are not fitted to the digit data, not defined in terms of the frequencies, and not justified solely by self-citation. The octal analogue follows the same pattern. No step reduces the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard facts from algebraic number theory such as the definition of multiplicative order and the existence of class numbers for imaginary quadratic fields; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Multiplicative order of 10 modulo p divides p-1 and is well-defined for prime p not dividing 10
    Invoked to define the period length as (p-1)/2 under the stated hypothesis.
  • standard math Class numbers of imaginary quadratic fields are well-defined positive integers
    Used as the target quantities in the frequency expressions.

pith-pipeline@v0.9.0 · 5618 in / 1380 out tokens · 65106 ms · 2026-05-18T09:05:48.529124+00:00 · methodology

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Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    B. C. Berndt, Classical theorems on quadratic residues, Enseign. Math. (2) 22 (1976), 261–304

    work page 1976

  2. [2]

    Chakraborty, K

    K. Chakraborty, K. Krishnamoorthy, On some symmetries of the basenexpansion of 1/m: the class number connection, Pacific J. Math. 319 (2022), 39–53

    work page 2022

  3. [3]

    Girstmair, The digits of 1/pin connection with class number factors, Acta Arith

    K. Girstmair, The digits of 1/pin connection with class number factors, Acta Arith. 67 (1994) 381–386

    work page 1994

  4. [4]

    Girstmair, Periodische Dezimalbr¨ uche – was nicht jeder dar¨ uber weiß, Jahrbuch ¨Uberblicke Mathematik 1995, Braunschweig, 1995, 163–179

    K. Girstmair, Periodische Dezimalbr¨ uche – was nicht jeder dar¨ uber weiß, Jahrbuch ¨Uberblicke Mathematik 1995, Braunschweig, 1995, 163–179

    work page 1995

  5. [5]

    Hasse, ¨Uber die Klassenzahl abelscher Zahlk¨ orper, Berlin, 1952

    H. Hasse, ¨Uber die Klassenzahl abelscher Zahlk¨ orper, Berlin, 1952

    work page 1952

  6. [6]

    Hirabayashi, Generalizations of Girstmair’s formula, Abh

    M. Hirabayashi, Generalizations of Girstmair’s formula, Abh. Math. Sem. Univ. Hamburg 75 (2005), 83–95

    work page 2005

  7. [7]

    Louboutin, Computation of class numbers of quadratic number fields, Math

    S. Louboutin, Computation of class numbers of quadratic number fields, Math. Comp. 71 (2002), 1735–1743

    work page 2002

  8. [8]

    Mizuno, A certain character twisted average value of the digits of rational num- bers and the class numbers of imaginary quadratic fields, Acta Arith

    Y. Mizuno, A certain character twisted average value of the digits of rational num- bers and the class numbers of imaginary quadratic fields, Acta Arith. 208 (2023), 215–233. 7

    work page 2023

  9. [9]

    Moree, On primes in arithmetic progression having a prescribed primitive root, J

    P. Moree, On primes in arithmetic progression having a prescribed primitive root, J. Number Th. 78 (1999), 85–98

    work page 1999

  10. [10]

    Moree, A

    P. Moree, A. Zumalac´ arregui, Salajan’s conjecture on discriminating terms in an exponential sequence, J. Number Th. 160 (2016), 646–665

    work page 2016

  11. [11]

    M. R. Murty, R. Thangadurai, The class number ofQ( √−p) and digits of 1/p, Proc. Amer. Math. Soc. 139 (2011), 1277–1289

    work page 2011

  12. [12]

    Pujahari, N

    S. Pujahari, N. Saikia,l-adic digits and class numbers of imaginary quadratic fields, Internat. J. Math. 35 (2024), Paper No. 2450041, 16 pp

    work page 2024

  13. [13]

    Shiomi, An analogue of Girstmair’s formula in function fields, Finite Fields Appl

    D. Shiomi, An analogue of Girstmair’s formula in function fields, Finite Fields Appl. 103 (2025), Paper No. 102585, 16 pp

    work page 2025

  14. [14]

    Szmidt, J

    J. Szmidt, J. Urbanowicz, D. Zagier, Congruences among generalized Bernoulli num- bers, Acta Arith. 71 (1995), 273–278

    work page 1995

  15. [15]

    M. H. Tom´ e, Arithmetic of Hecke L-functions of quadratic extensions of totally real fields, J. Number Th. 262 (2024), 103–133

    work page 2024

  16. [16]

    L. C. Washington, Introduction to Cyclotomic Fields. New York, 1982. Kurt Girstmair Institut f¨ ur Mathematik Universit¨ at Innsbruck Technikerstr. 13/7 A-6020 Innsbruck, Austria Kurt.Girstmair@uibk.ac.at 8

    work page 1982