bibkey: somerkrizek2025uniformsubsequence authors: Lawrence Somer; Michal Krizek year: 2025 title: “Generalization of a Theorem of Velez on Uniform Distribution in Second-Order Linear Recurrences” doi: 10.5281/zenodo.14679256 url: https://math.colgate.edu/~integers/z1/z1.pdf claim: “Conjecture 4.2 proposes that every uniformly distributed k-th order integer recurrence modulo p^e, with least period p^e E and last coefficient coprime to p, remains uniformly distributed on every coprime-step subsequence, with E/gcd(g,E) copies of each residue in the stated finite range.” strata_touched:

• D5/S1/Recurrence/Periodic/SomerUniformSubsequenceRefutation license: citation-only triage: anchor

Somer and Krizek, Conjecture 4.2

The paper defines the k-th order integer recurrence

w(n+k) = a1*w(n+k-1) + a2*w(n+k-2) + ... + ak*w(n)

for k >= 2. Conjecture 4.2 assumes that modulo p^e the sequence has least period p^e E, that the last coefficient ak is coprime to p, and that every residue occurs exactly E times in that least period. For each positive g coprime to p and nonnegative s, it sets r = E/gcd(g,E) and asserts that every residue occurs exactly r times among w(s+ng) for 0 <= n < p^e r.

The paper’s Example 4.3 supplies positive evidence from a different third-order recurrence. It does not prove the conjecture.

Verified locator

• DOI: 10.5281/zenodo.14679256
• URL: https://math.colgate.edu/~integers/z1/z1.pdf
• Printed location: page 8, Conjecture 4.2
• Publication: INTEGERS 25 (2025), A1, published January 17, 2025