Let varphi=(1+√5)/2. For every admissible terminal block word w occurring in canonical Zeckendorf representations, if its total digit-length is m, the conjecture asserts that lim_X→∞frac#p≤ X: p is prime and the Zeckendorf representation of p terminates in wpi(X)=varphi^-m. For the Tribonacci substitution fixed word u, and every finite factor w of u, the corresponding conjecture asserts that lim_X→∞frac#p≤ X: p is prime and u_pu_p+1… u_p+|w|-1=wpi(X)=operatornamefreq_u(w), where operatornamefreq_u(w) is the ordinary factor frequency of w in u.

Zeckendorf terminal-block conjecture and Tribonacci prime-number conjecture

Let $\varphi=(1+\sqrt{5})/2$ . For every admissible terminal block word $w$ occurring in canonical Zeckendorf representations, if its total digit-length is $m$ , the conjecture asserts that

\lim_{X\to\infty}\frac{\#\{p\le X:\ p\text{ is prime and the Zeckendorf representation of }p\text{ terminates in }w\}}{\pi(X)}=\varphi^{-m}.

For the Tribonacci substitution fixed word $u$ , and every finite factor $w$ of $u$ , the corresponding conjecture asserts that

\lim_{X\to\infty}\frac{\#\{p\le X:\ p\text{ is prime and }u_{p}u_{p+1}\cdots u_{p+|w|-1}=w\}}{\pi(X)}=\operatorname{freq}_{u}(w),

where $\operatorname{freq}_{u}(w)$ is the ordinary factor frequency of $w$ in $u$ .

References

Primary source

arXiv

Additional references

Terminal Blocks of Primes in Pisot Numeration Systems — arXiv — Sungkon Chang, Johann Verwee

Posted 8 hours ago · edited 8 hours ago

Progress summary

Refreshed 8 hours ago

Claimed solved

A new preprint claims to settle both conjectures by describing how prime numbers distribute through these special number representations, but the claim has not been independently verified.

The problem asks whether prime numbers exhibit the predicted terminal-block and substitution-pattern frequencies in Zeckendorf and Tribonacci representations. Sungkon Chang and Johann Verwee claim a broader Pisot-system theorem that would settle both conjectures.

Known results

A 2022 study of the Tribonacci component reports that the Marques–Lengyel conjecture fails for infinitely many primes $p \equiv 2 \pmod 3$ , and for every $p \in [5,599]\setminus\{11,83,103,163,397\}$ in the stated range.

October 2026 preprint

Chang and Verwee claim that fixed terminal words occur among primes with predicted frequencies, while substitution factors occur at prime positions with ordinary frequencies. If correct, this settles the tracked Zeckendorf and Tribonacci conjectures; the result is presently unverified.

Current status (as of October 2026): Chang and Verwee claim both conjectures are settled, but the claim remains unverified and no independent assessment was found.

Sources

No solutions have been posted yet.