On the normality of the concatenated Fibonacci constant

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 $F_{n}$.

• The paper establishes that the concatenated Fibonacci constant’s normality hinges on the (ε, k)-normality of nearly all Fibonacci numbers.
• It identifies key obstructions arising from exponential growth, Benford bias, and Pisano periodicity that challenge traditional normality proofs.
• Extensive numerical experiments in bases 10 and 2 demonstrate statistical uniformity, supporting the sufficiency criterion for normality despite theoretical limits.

On the Normality of the Concatenated Fibonacci Constant

Introduction

The paper "On the normality of the concatenated Fibonacci constant" (2604.17136) presents a rigorous analysis of the normality problem for the number formed by concatenating decimal (or binary) representations of the Fibonacci numbers. Specifically, the concatenated Fibonacci constant is defined as $\mathcal{F} = 0.F_1F_2F_3\cdots = 0.11235813...$, where $F_n$ denotes the $n$th Fibonacci number. The principal question addressed is whether $\mathcal{F}$ is normal in a given base, i.e., whether every finite block of digits occurs with the expected frequency in the constant’s base-$b$ expansion.

This problem is situated at the intersection of combinatorial number theory, uniform distribution, and probabilistic digit phenomena. The Fibonacci sequence is a prototypical example of a simple linear recurrence with exponential growth. The paper utilizes both structural analysis of digit distributions and large-scale numerical experimentation to investigate the normality problem, identifying the precise obstructions to existing theoretical tools.

Limitations of Classical Criteria

Classical results on normal numbers constructed via concatenation—for example, the Champernowne constant, the Besicovitch number, and the Copeland–Erdős constant—explain normality for sequences whose growth is at most polynomial. These rely on the count of sequence elements up to $N$ being sufficiently dense among the positive integers and on the digits-per-term growing slowly (e.g., as $O(n^{1-\delta})$).

The Fibonacci sequence grows as $F_n \sim \phi^n/\sqrt{5}$ (with $\phi$ the golden ratio), and the counting function $\Phi_F(N)$ for how many Fibonacci numbers are $F_n$0 is only $F_n$1. This exponential sparsity fails the density and digit-length conditions of Copeland–Erdős and Davenport–Erdős, and the sequence is not covered by more modern polynomial or generalized polynomial frameworks. Similarly, constructions based on iterated totient, sum-of-divisors, or Carmichael’s lambda functions (e.g., the Pollack–Vandehey $F_n$2–$F_n$3–$F_n$4 class) fail, since no composition of these functions grows exponentially, nor captures the precise arithmetic of the Fibonacci sequence.

The paper formally proves that the concatenated Fibonacci constant lies beyond the reach of all existing concatenation normality theorems and that the classical machinery fundamentally cannot settle its normality.

Sufficient Condition via $F_n$5-Normality

The central theoretical contribution is the isolation of a sharp sufficient condition for normality, resulting from the normality criterion of Pollack–Vandehey. Specifically, it is shown that if almost all Fibonacci numbers are $F_n$6-normal in base $F_n$7 (i.e., their sliding blocks of length $F_n$8 approximate uniform frequency within $F_n$9 for all $n$0), then the concatenated constant $n$1 will be normal.

Formally: if for every $n$2 and $n$3, the proportion of $n$4 such that $n$5 is not $n$6-normal is $n$7, then $n$8 is base-$n$9 normal. The authors verify conditions for the average and maximum digit length growth required in this scheme and reduce the problem to $\mathcal{F}$0-normality of the individual $\mathcal{F}$1.

This focus isolates digit normality within most Fibonacci numbers in the relevant base as the key obstacle.

Structural Sources of Digit Bias

Two possible asymptotic sources of bias in the digit distribution of $\mathcal{F}$2 are separately quantified:

1. Benford (Leading Digit) Bias: Leading digits of Fibonacci numbers follow Benford's law due to the equidistribution of the fractional parts of $\mathcal{F}$3, with the measure for leading digit $\mathcal{F}$4 as $\mathcal{F}$5. However, these contribute only $\mathcal{F}$6 of the total digits up to the $\mathcal{F}$7th Fibonacci number, and their effect vanishes asymptotically.
2. Pisano (Trailing Digit) Periodicity: Trailing digits are governed by the periodicity of the Fibonacci sequence modulo powers of the base (Pisano periods). Again, only $\mathcal{F}$8 of digits are so affected, and asymptotically these play a negligible role in the total digit counts.

The deep (middle) digits remain, which are not governed by such explicit periodic or logarithmic phenomena.

Deep Digit Distribution and Failure of Discrepancy Machinery

The core remaining obstruction comes from the distribution of "deep" digits—those whose positions in $\mathcal{F}$9 are neither leading nor trailing, but instead proportional to the length of $b$0.

Attempts to control their distribution by Koksma–Hlawka-type discrepancy bounds with bounded variation fail, due to exponential sensitivity: the relevant indicator functions for fixed digit values at deep positions have variation of the same exponential magnitude as $b$1 itself, leading to divergence in the error terms.

Thus, even under strong Diophantine conditions (e.g., badly approximable parameters), current uniform distribution and discrepancy theoretic tools cannot pin down the deep digits, precluding a theoretical proof under the present techniques.

Benfield–Manes Row vs. Column Analysis

The paper gives a rigorous critique of the proposal by Benfield and Manes, who showed that, in certain bases (products of powers of $b$2 and $b$3), the digits at each fixed position modulo the base period in the Fibonacci sequence achieve perfect uniformity across one Pisano period. The authors clarify that this is a statement about row-wise (position-fixed, sequence-varied) uniformity in the two-dimensional digit array indexed by $b$4, while normality in concatenation refers to column-wise (number-fixed, position-varied) uniformity. The logical independence of these two types of equidistribution is highlighted via explicit counterexamples.

Crucially, the column-wise uniformity required for normality is not implied by (and does not follow from) the per-position Pisano uniformity; thus, the result of Benfield–Manes does not resolve the underlying normality question.

Large-Scale Numerical Experiments

Extensive computational experiments are conducted on the first $b$5 Fibonacci numbers in bases $b$6 and $b$7, yielding more than $b$8 decimal digits and over $b$9 binary digits respectively.

• Global Digit Frequencies: For both bases, single-digit frequencies across all digits are statistically indistinguishable from the expected uniform values, with maximal deviations decaying as $N$0 in the number of digits $N$1.
• Block Frequencies: For $N$2, block (substring) frequencies closely match the uniform expectation, with chi-squared statistics and Good's serial test showing compatibility with independence and uniformity at the tested scales.
• Per-$N$3 $N$4-Normality: The proportion of Fibonacci numbers failing $N$5-normality for various $N$6 is small and vanishing, providing empirical support (though not a proof) for the sufficiency criterion.
• Positional Analysis: Structured deviations are concentrated at the boundaries between concatenated Fibonacci numbers (due to Benford and Pisano effects), but their contribution is asymptotically negligible. The pooled interior (middle) digits and blocks are essentially uniform.

No persistent bias or anomaly is observed that would suggest deviation from normality.

Implications and Prospects

This work demonstrates numerically that, at all computationally accessible scales, the concatenated Fibonacci constant behaves as expected of a normal number in both base 10 and 2. Theoretical obstacles are localized and precisely identified: the only plausible obstruction to normality is the unproven assertion that the base-$N$7 digits of most large $N$8 are equidistributed at all finite block lengths.

The inability of existing discrepancy/equidistribution methods to resolve the deep digit behavior is paralleled by similar open problems in exponential sequences, such as the conjecture that $N$9 is normal for most $O(n^{1-\delta})$0—a notoriously hard question.

Advancement may require new techniques, possibly using modern ideas from dynamical systems, probabilistic combinatorics of iterated recurrences, or higher-order mixing induced by deterministic carries. The paper suggests potential connections to mixing properties in additive processes and the study of finite-state dimension as partial relaxations of normality.

Conclusion

"On the normality of the concatenated Fibonacci constant" (2604.17136) provides a definitive account of known obstacles and empirical evidence for the normality of the number $O(n^{1-\delta})$1 in various bases. Theoretical results isolate the equidistribution properties of deep digits of $O(n^{1-\delta})$2 as the only substantial obstruction and demonstrate the insufficiency of all current discrepancy-based methods. Empirical data from concatenations of hundreds of thousands of Fibonacci numbers reveal no significant deviation from normality at observable scales, with convergence rates in line with random models. The question of normality remains open in all bases, but this work exhaustively identifies where progress must occur and provides all available computational support for the conjectured normality. Future work in this domain must confront the control of exponential recurrence digit distributions in settings unamenable to existing tools.