On the number of residues of linear recurrences

On Two New Identities Concerning the Q-Integers and Several Divisibility Properties for These Integers

“…Quantum calculus or q -calculus receiving significant attention from researchers play an important role in various applications in sciences and technology (see, [1][2][3][4][5][6]). Several books already present general survey for quantum calculus as [7][8][9][10] for the more recent ones.…”

Limiting density of the Fibonacci sequence modulo powers of $p$

“…Burr [3] completed the characterization of integers m ≥ 2 for which this density is 1 (that is, every residue is attained): The sequence (F (n) mod m) n≥0 contains all residues modulo m if and only if m = 5 k m for some k ≥ 0 and some m ∈ {2, 4, 6, 7, 14} ∪ {3 j : j ≥ 0}. More recently, Dubickas and Novikas [4] showed that for every k ≥ 1 there exists a modulus m and a sequence satisfying the Fibonacci recurrence that attains exactly k residues modulo m. Sanna [8] studied this question for other secondorder recurrences. For a constant-recursive sequence satisfying a general secondorder recurrence, Bumby [2] characterized the moduli for which the residues are uniformly distributed.…”