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arxiv: 2604.17136 · v1 · submitted 2026-04-18 · 🧮 math.NT · math.PR· math.ST· stat.TH

On the normality of the concatenated Fibonacci constant

Jos\'e Ricardo G. Mendon\c{c}a This is my paper

Pith reviewed 2026-05-10 05:58 UTC · model grok-4.3

classification 🧮 math.NT math.PRmath.STstat.TH
keywords normalityFibonacci sequenceconcatenated constantdigit distributionBenford lawPisano periodinterior digits
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The pith

The concatenated Fibonacci constant may be normal in base 10 if interior digits of large Fibonacci numbers remain uniform.

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

The paper examines the number formed by writing the Fibonacci sequence as a decimal 0.1123581321..., and asks whether this concatenated constant is normal. Standard sufficient conditions for normality fail here because the Fibonacci numbers grow exponentially fast. A different criterion reduces normality of the whole to the requirement that almost every individual Fibonacci number has nearly uniform digits. Computations through the first 500000 terms show that leading digits obey Benford's law and trailing digits repeat with the Pisano period, yet both occupy a vanishing share of all digits produced. Interior blocks of digits, once the boundaries are removed, match the frequencies expected from a uniform source at the scales checked.

Core claim

The concatenated Fibonacci constant F is obtained by stringing together the decimal expansions of F1, F2, F3, … . Because of exponential growth, classical concatenation tests do not apply, but a criterion of Pollack and Vandehey shows that normality would follow if almost all Fn are (ε,k)-normal. The Benford bias at the left end and the Pisano periodicity at the right end each contribute o(1) fractions of the total digit count. Large-scale checks on the first 500000 terms find that pooled interior k-blocks for k=2,3,4 stay statistically consistent with uniformity, while visible structure concentrates only at the junctions between consecutive numbers. The computations therefore locate any non

What carries the argument

Positional decomposition of each Fibonacci number into leading, interior, and trailing segments, isolating the vanishing boundary contributions from the bulk digit statistics that control normality.

If this is right

  • If interior digit blocks stay uniform at all scales, the concatenated constant is normal in base 10.
  • The exponential growth of Fibonacci numbers makes the fractional contribution of leading and trailing digits tend to zero.
  • Global single-digit and short-block counts in the first 500000 terms fluctuate exactly as they would for an iid uniform source.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same boundary-versus-interior separation may decide normality for concatenations of other exponentially growing integer sequences.
  • If the deep digits of Fibonacci numbers are asymptotically normal, the construction supplies an explicit recursive example of a normal number.
  • Extending the same positional analysis to bases other than 10 or 2 would test whether the observed uniformity is base-dependent.

Load-bearing premise

The uniformity seen in interior digit blocks for the first 500000 Fibonacci numbers continues without systematic drift for all larger terms.

What would settle it

A statistically significant deviation from uniform frequencies in the interior blocks of Fibonacci numbers larger than the 500000th term would show that the pattern does not hold.

read the original abstract

We study the concatenated Fibonacci constant $\mathcal{F} := 0.F_{1}F_{2}F_{3}\cdots = 0.11235813\cdots$, obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical sufficient conditions for normality by concatenation do not apply to the Fibonacci sequence because of its exponential growth, while a criterion of Pollack and Vandehey implies that the normality of $\mathcal{F}$ in base $10$ would follow if almost all Fibonacci numbers were $(\varepsilon,k)$-normal in base $10$. The Benford bias of leading digits and the Pisano periodicity of trailing digits are shown to contribute asymptotically negligible fractions of the total digits, isolating the distribution of the deep digits of large Fibonacci numbers as the remaining obstruction. Large-scale numerical experiments on the first $500{,}000$ Fibonacci numbers in bases $10$ and $2$ indicate that global single-digit counts and $k$-block statistics for $k = 2, 3, 4$ are compatible with iid-like fluctuations at the scales tested, and that a positional decomposition concentrates the visible structured deviation at the boundaries between consecutive Fibonacci numbers, while pooled interior blocks remain close to uniform. Our computations suggest that any obstruction to normality lies in the asymptotic behavior of the deep digits of $F_{n}$.

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 / 3 minor

Summary. The manuscript examines the concatenated Fibonacci constant F := 0.F1 F2 F3 ⋯ and asks whether it is normal in base 10. It demonstrates that classical sufficient conditions for normality of concatenated sequences fail because of the exponential growth in the number of digits of Fn. A criterion of Pollack and Vandehey is applied to reduce the question to the (ε,k)-normality of almost all individual Fibonacci numbers. Leading-digit Benford bias and trailing-digit Pisano periodicity are shown to contribute only o(1) fractions of the total digits. Numerical experiments on the first 500,000 Fibonacci numbers in bases 10 and 2 report that global single-digit frequencies and pooled interior k-block counts (k ≤ 4) remain statistically compatible with uniformity, with visible deviations localized at the boundaries between consecutive terms. The authors conclude that any obstruction to normality must therefore lie in the asymptotic interior-digit behavior of large Fn.

Significance. If the reduction via the Pollack-Vandehey criterion is valid, the work usefully isolates the normality question for F to the interior digits of Fibonacci numbers, ruling out boundary effects at the scales examined. The theoretical arguments correctly explain the inapplicability of earlier concatenation criteria and the asymptotic negligibility of leading/trailing biases. The large-scale numerical evidence, while finite and suggestive rather than conclusive, is consistent with the proposed picture and provides a concrete benchmark for future investigations of digit distributions in Fibonacci numbers.

major comments (2)
  1. [§3] §3 (Pollack-Vandehey reduction): the manuscript invokes the criterion to conclude that normality of F follows from (ε,k)-normality of almost all Fn, but does not state the precise quantitative form of the criterion (including the required uniformity in ε and the dependence on k) or verify that the exceptional set has density zero in the sense needed for the concatenation measure. This step is load-bearing for the reduction claim.
  2. [§5] §5 (numerical experiments): the pooled interior k-block frequencies for k=2,3,4 are described as 'compatible with iid-like fluctuations,' yet no explicit discrepancy measure, chi-squared statistic, or p-value is reported for the 500,000-term data set. Without these, it is difficult to quantify how close the observed counts are to uniform and whether the compatibility persists beyond the tested scale.
minor comments (3)
  1. The phrase 'deep digits' appears in the abstract and conclusion but is never given a precise definition (e.g., digits excluding the leading O(log n) and trailing O(1) blocks). A short clarifying sentence would improve readability.
  2. The abstract states that results hold 'in bases 10 and 2' but provides no summary statistics or figures for base 2; a single sentence or table entry comparing the two bases would make the claim easier to assess.
  3. Notation for the concatenated constant uses script F; ensure that this symbol is introduced once in the introduction and used consistently thereafter, avoiding any overlap with the Fibonacci sequence symbol F_n.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help clarify the presentation of the Pollack-Vandehey reduction and strengthen the numerical evidence. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Pollack-Vandehey reduction): the manuscript invokes the criterion to conclude that normality of F follows from (ε,k)-normality of almost all Fn, but does not state the precise quantitative form of the criterion (including the required uniformity in ε and the dependence on k) or verify that the exceptional set has density zero in the sense needed for the concatenation measure. This step is load-bearing for the reduction claim.

    Authors: We agree that the precise quantitative form of the Pollack-Vandehey criterion, including uniformity in ε and dependence on k, was not stated explicitly, nor was the density-zero condition for the exceptional set verified in detail. In the revised manuscript we will quote the full statement of the criterion and confirm that the exceptional set of Fibonacci numbers failing (ε,k)-normality has density zero with respect to the concatenation measure, thereby making the reduction rigorous. revision: yes

  2. Referee: [§5] §5 (numerical experiments): the pooled interior k-block frequencies for k=2,3,4 are described as 'compatible with iid-like fluctuations,' yet no explicit discrepancy measure, chi-squared statistic, or p-value is reported for the 500,000-term data set. Without these, it is difficult to quantify how close the observed counts are to uniform and whether the compatibility persists beyond the tested scale.

    Authors: The referee is correct that the manuscript provides only a qualitative description of compatibility without reporting explicit statistics such as chi-squared values or p-values. We will add these quantitative measures for the pooled interior k-block counts (k=2,3,4) in the revised version, computed directly from the 500,000-term dataset, to allow precise assessment of the observed uniformity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper invokes the external Pollack-Vandehey criterion to reduce the normality question to almost-all individual F_n being (ε,k)-normal; this is an independent theorem, not derived within the paper. Asymptotic negligibility of Benford leading-digit bias and Pisano trailing-digit periodicity is established by direct estimates on the total digit count contributed by boundaries (o(1) fraction), without fitting parameters or self-referential definitions. The reported k-block frequency statistics (k≤4) are raw pooled counts from explicit computation on the first 500000 terms, presented as empirical evidence rather than predictions from a fitted model. No self-citations are load-bearing for the central reduction or conclusion; the final suggestion that any obstruction lies in deep-digit asymptotics follows from the separation of boundary effects and the external criterion, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts about Fibonacci growth, Pisano periods, and existing normality criteria; no new free parameters, axioms, or invented entities are introduced.

axioms (2)
  • standard math Fibonacci numbers grow exponentially and possess Pisano periods for trailing digits
    Invoked to show that Benford bias and trailing periodicity contribute negligibly many digits.
  • standard math Pollack-Vandehey criterion for normality by concatenation
    Used to reduce the problem to almost-all Fibonacci numbers being (ε,k)-normal.

pith-pipeline@v0.9.0 · 5548 in / 1349 out tokens · 52984 ms · 2026-05-10T05:58:01.021189+00:00 · methodology

discussion (0)

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Reference graph

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