The conjectured first-difference sequence for the Fibonacci word

Less than 1 year old · traced to 2026

Let $f$ be the infinite Fibonacci word, let $\Sigma$ be its alphabet, and define

M_n=\min\{\operatorname{Cl}(w):w\in\operatorname{Fac}(f)\cap\Sigma^n\},\qquad R_n=M_n-M_{n-1}.

Here $\operatorname{Cl}(w)$ is the number of distinct closed factors of $w$ , and $F_n$ denotes the Fibonacci sequence used in the source. Fibonacci first-difference conjecture. The sequence $(R_n)_{n\geq1}$ is

1,1,1,1,F_0,F_2,F_0,F_2,F_1,F_1,F_3,F_1,F_1,F_3,F_2,F_2,F_2,F_4,F_4,F_2,F_2,F_2,F_4,F_4,\ldots,

with the displayed infinite product pattern specified in the source.

The conjecture is based on numerical experiments and would yield an explicit formula for $M_n$ . The paper does not report a proof or a resolution.

References

Primary source

Anuran Maity and Svetlana Puzynina, “Bounds on the closed-rich constant of infinite words”, arXiv:2605.19535 (2026).

combinatorics-on-words fibonacci-word

Posted 1 month ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A reader-submitted proof claims to settle the conjecture, but no independent verification has been found.

The conjecture predicts an exact repeating pattern for the first differences $R_n=M_n-M_{n-1}$ , and therefore an explicit formula for the minimum number of distinct closed factors in length- $n$ factors of the Fibonacci word.

May 2026 numerical conjecture

The paper presents the pattern from numerical experiments and reports no proof or resolution.

Community submission (unverified), September 29, 2026

A submitted proof argues that singular-kernel analysis gives exact minimizers on every Fibonacci interval, proves the stated first-difference product, identifies unique minimizers at $N_k=F_k+2F_{k-3}-2$ , and derives an associated limiting ratio. These claims have not been independently verified.

Current status (as of September 2026): the conjecture has an unverified submitted proof claim, but no independently confirmed proof or counterexample is recorded.

Sources

ProofI prove Conjectures 5.2 and 5.3 of Maity–Puzynina for the Fibonacci word. The argument determines the exact minimum M_n of distinct closed factors for every n, hence the conjectured first-difference sequence and the exact closed-rich constant C_f=φ^3/(φ^3+2)^2. It also proves the conjectured uniqueness of the minimizing factor at n=F_k+2F_{k-3}-2 for k≥5.See full solutionHide full solution

Let $f$ be the infinite Fibonacci word and define

M_n=\min\{\operatorname{Cl}(w):w\in\operatorname{Fac}(f),\ |w|=n\},

where $\operatorname{Cl}(w)$ denotes the number of distinct closed factors of $w$ , including the empty word.

The manuscript determines $M_n$ exactly for every $n$ . In particular, taking first differences

R_n=M_n-M_{n-1}

gives

(R_n)_{n\ge1} = (1,1,1,1) \prod_{j=0}^{\infty} \left( F_j^{[F_j]}, F_{j+2}^{[F_{j-1}]}, F_j^{[F_j]}, F_{j+2}^{[F_{j-1}]} \right),

where $v^{[r]}$ denotes $r$ consecutive copies of $v$ . This proves Conjecture 5.2.

The proof starts from the singular-kernel description of factors of the Fibonacci word. Singular kernels give explicit intervals for suffixes ending at a fixed position, and the preceding occurrence of the same kernel determines exactly which of these suffixes repeat inside a given factor. This yields explicit repeated-suffix data for the three possible singular-kernel families.

For a fixed length, write $G(i)$ for the number of closed factors in a member of one such family and $r(i)$ for the corresponding repeated-suffix increment. Reversal symmetry gives

G(i)=G(h-i)

and

G(i+1)-G(i)=r(h-i-1)-r(i).

The function $r$ has only finitely many downward jumps in each family. Consequently $G$ is discretely concave between the resulting candidate indices, so its minimum is attained at one of these explicitly determined points. Evaluating and comparing those candidates gives the exact piecewise formula for $M_n$ on every Fibonacci interval

[F_k,F_{k+1}-1].

N_k=F_k+2F_{k-3}-2,\qquad k\ge5,

the minimizing interval collapses to a single parameter value. Hence the minimizing factor is unique and equals

f_{k-3}f_kf_{k-3}^{--},

with

M_{N_k}=1+F_{k-3}F_k.

Finally, the exact formula for $M_n$ reduces the minimization of $M_n/n^2$ to the endpoints between successive changes of slope. The relevant competing values occur at

N_k=F_k+2F_{k-3}-2

and at the other upward-slope breakpoints. Using the exact Fibonacci/Binet relation and passing to the limit gives

C_f = \inf_{n\ge1}\frac{M_n}{n^2} = \frac{\varphi^3}{(\varphi^3+2)^2}, \qquad \varphi=\frac{1+\sqrt5}{2}.

Thus Conjecture 5.3 also holds.

A full proof, including the suffix-interval calculations, the finite initial cases, and all comparisons required in the minimization, is contained in the attached manuscript. The LaTeX source is publicly available in the GitHub repository FDmd233/fibonacci-closed-factor-minima.

The manuscript is presented for independent verification and has not yet undergone journal peer review.

paper_en.pdf399,592 bytesOpen