Universal Modular Realization of Digit Blocks in k-Bonacci Sequences
Year: 2026
Type: article
Abstract: This paper establishes the disjunctivity of concatenations formed from nontrivial k-bonacci sequences with nonnegative integer initial values, for every recurrence order k ≥ 2 and every integer base b ≥ 2. In particular, every finite digit word occurs infinitely often in the resulting concatenation. Two independent proofs are presented. The first develops a recursive modular approach using p-adic analytic interpolation, local bijections, and the Chinese remainder theorem. It establishes a stronger realization result: along a suitable arithmetic progression of indices, prescribed digit blocks occur at fixed positions with exact frequencies over complete modular windows. The second proof follows the logarithmic equidistribution method associated with Benford's law and Washington's work on Fibonacci numbers. By extending the argument to general k-bonacci sequences and arbitrary leading digit words, it provides an independent route to disjunctivity. The two approaches illuminate complementary aspects of digit occurrence: modular realization at prescribed positions and asymptotic distribution of leading digit words. The results distinguish disjunctivity from normality; they do not establish that the concatenations are normal.
Author: Eigler, Dieter
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225320369