Let F_m be the set of periodic sequences from the Fibonacci recurrence or its parity transform with initial condition (a_0,a_1)∈( Z/m Z)^2, taken modulo m. For a divisor d of a composite modulus m, define the weight of a period q of length ell_q in F_d by w_d(q)=fracell_qd^2. If q_d∈ F_d extends to periods q_m^(1),…,q_m^(k)∈ F_m reducing to q_d modulo d, Weight Preservation of Fibonacci Recurrences. w_d(q_d)=∑_i=1^k w_m(q_m^(i)). Thus the total weight of every period is conjectured to be conserved under extension from d to m; the supplied text gives examples but no resolution status.

Weight preservation for Fibonacci recurrences

About 1 year old · traced to 2025

Let $F_m$ be the set of periodic sequences from the Fibonacci recurrence or its parity transform with initial condition $(a_0,a_1)\in(\mathbb Z/m\mathbb Z)^2$ , taken modulo $m$ . For a divisor $d$ of a composite modulus $m$ , define the weight of a period $q$ of length $\ell_q$ in $F_d$ by

w_d(q)=\frac{\ell_q}{d^2}.

If $q_d\in F_d$ extends to periods $q_m^{(1)},\ldots,q_m^{(k)}\in F_m$ reducing to $q_d$ modulo $d$ , Weight Preservation of Fibonacci Recurrences.

w_d(q_d)=\sum_{i=1}^k w_m(q_m^{(i)}).

Thus the total weight of every period is conjectured to be conserved under extension from $d$ to $m$ ; the supplied text gives examples but no resolution status.

References

Primary source

Marc T. Pudelko, “Modular Periodicity of Random Initialized Recurrences”, arXiv:2510.24882 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

Computations support the conjecture, while an unverified posted argument claims a general counting proof.

Marc T. Pudelko formulated the conjecture in a preprint first posted in October 2025. It says that the normalized total length of every period is preserved when Fibonacci or parity-transformed sequences are lifted from a divisor modulus to a composite modulus.

October 2025 computational evidence

The preprint checks exact preservation in examples including lifts from $d=2$ to $m=6$ , from $d=3$ to $m=6$ , and prime-power cases such as $19$ to $361$ . It gives no general proof, counterexample, withdrawal, or independent verification.

Posted attempt

A posted argument claims a complete proof by counting fibers of the reduction map between finite orbit sets: each fiber has size $(m/d)^2$ , and decomposing the inverse image of an orbit yields the asserted weight identity. This attempt has not been independently verified.

Current status (as of August 2026): The conjecture has an unverified complete proof attempt, but no verified proof or counterexample is recorded, so its mathematical resolution remains open.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

The claim follows from an orbit-fiber counting identity, valid much more generally than the Fibonacci recurrence.

For $\varepsilon\in\{1,-1\}$ , let

A_\varepsilon= \begin{pmatrix}0&1\\1&\varepsilon\end{pmatrix}.

The Fibonacci recurrence and its parity transform act on initial-condition pairs by

(a_j,a_{j+1}) \longmapsto (a_{j+1},a_j+\varepsilon a_{j+1}) =A_\varepsilon(a_j,a_{j+1}).

Since $\det A_\varepsilon=-1$ , this is a permutation of

X_s=(\mathbb Z/s\mathbb Z)^2

for every $s\ge1$ . Its permutation orbits are precisely the periods modulo $s$ , and orbit cardinality equals period length.

For $d\mid m$ , coordinatewise reduction

\pi:X_m\longrightarrow X_d

commutes with both recurrence actions:

\pi A_{\varepsilon,m}=A_{\varepsilon,d}\pi.

Every fiber has cardinality

|\pi^{-1}(x)|=(m/d)^2.

Fix an orbit $O\subset X_d$ of length $\ell$ . Its full inverse image is invariant under $A_{\varepsilon,m}$ , so it decomposes into disjoint orbits

\pi^{-1}(O)=O_1\sqcup\cdots\sqcup O_r.

Each $O_i$ maps onto $O$ , since its image is a nonempty invariant subset of a single transitive orbit. Thus the $O_i$ are exactly all periods modulo $m$ reducing to the chosen period modulo $d$ .

Counting the inverse image yields

\sum_{i=1}^r|O_i| = |\pi^{-1}(O)| = \ell\left(\frac md\right)^2.

Dividing by $m^2$ gives precisely

\sum_{i=1}^r w_m(O_i) = \sum_{i=1}^r\frac{|O_i|}{m^2} = \frac{\ell}{d^2} = w_d(O).

More generally, if $A$ is any integral $r\times r$ matrix whose determinant is coprime to $m$ , then for every $d\mid m$ and every orbit $O$ modulo $d$ ,

\sum_{\substack{O'\text{ orbit modulo }m\O'\bmod d=O}} \frac{|O'|}{m^r} = \frac{|O|}{d^r}.

Thus orbit-weight preservation holds for every invertible integral linear recurrence in every dimension.