Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

bibkey: shi2026second authors: Minjia Shi and Xuan Wang and Bouazzaoui Zakariae and Jon-Lark Kim and Patrick Sole year: 2026 title: Second order Recurrences, quadratic number fields and cyclic codes doi: 10.48550/arXiv.2603.25343 claim: Equality of the Pisano period modulo p and modulo p squared corresponds to a failure of p-rationality of the associated real quadratic field. strata_touched:

  • D5/S1/Scale/Fibonacci
  • D5/S3/Arith/FibonacciRank
  • D5/S3/Arith/GoldenApparition
  • D5/S3/Arith/GoldenPrimeSplitting license: citation-only triage: anchor

Second order Recurrences, quadratic number fields and cyclic codes

Shi, Wang, Zakariae, Kim, and Sole relate second order recurrences to quadratic number fields and cyclic codes. The paper restates Wall’s question — whether there is a prime p with pi(p) = pi(p^2) for the Pisano period pi — records that no such prime is known below 10^14, and cites heuristics conjecturing infinitely many. For its generalized recurrences it makes the connection precise: equality of the periods modulo p and p^2 corresponds, under stated hypotheses, to failure of p-rationality of the associated real quadratic field.

This note is the literature anchor for the problem candidate Problems/wall-sun-sun-golden-unit-lift.md.

Search log

  • 2026-08-18: Queried the arXiv Atom API for id_list=2603.25343. HTTP 200 with totalResults=1; the entry resolved to http://arxiv.org/abs/2603.25343v1, title Second order Recurrences, quadratic number fields and cyclic codes, authors Minjia Shi, Xuan Wang, Bouazzaoui Zakariae, Jon-Lark Kim, and Patrick Sole, published 2026-03-26, primary category math.NT. The API reported no arxiv:doi and no arxiv:journal_ref, so the arXiv-assigned DOI is used. The final author’s surname carries an acute accent in the API response; this note records it unaccented because the metadata line is restricted to ASCII.
  • 2026-08-18: Issued HEAD https://doi.org/10.48550/arXiv.2603.25343, which returned HTTP 302 redirecting to https://arxiv.org/abs/2603.25343.

Wall’s question is a long-standing open problem restated by this paper rather than originated by it; no search was performed for a resolution after this arXiv version.

Verified locator

  • arXiv: https://arxiv.org/abs/2603.25343
  • DOI: https://doi.org/10.48550/arXiv.2603.25343