We study the canonical normalization map behind Zeckendorf addition: a finite word over {0,1, 2} is replaced by the unique admissible Zeckendorf word with the same Fibonacci value. We first present this map as the canonical normalization induced by a section of the Fibonacci congruence on finite-support digit-count sequences and give a terminating local rewrite presentation whose terminal irreducible word is independent of the rewrite strategy. The main results give exact least-significant-digit-first (LSDF) noncausality invariants for this canonical map. For every output coordinate j , the LSDF anticipation horizon is infinite, even after restricting the domain to genuine digitwise sums of two admissible Zeckendorf inputs: A { 0,1,2 }(ℕ) ( j ) = A Add 2 ( j ) = ∞. The map has no continuous extension at the natural alternating one-sided limit tails. For LSDF finite words, the online commitment deficit is maximal: B ( n ) = n ; equivalently, the prefix-destruction index – the largest number of initial normalized-output digits that may be invalidated by extending an input – satisfies Δ( n ) = n . The same maximal prefix erasure holds on the genuine addition image, Δ Add 2 ( n ) = n , and the input-scale reduced residual complexity satisfies R in ( n ) ≥ n . Non-subsequentiality – that is, impossibility of realization by a deterministic one-pass finite transducer with bounded terminal output – is then a corollary of these sharper quantitative obstructions. These statements are specific to the canonical greedy Zeckendorf normalizer in the LSDF one-sided model and are not claims of a new general normalization theory for Fibonacci or confluent numeration systems. We contrast them with the classical most-significant-digit-first Berstel adder. The existence and ten-state presentation of that adder are classical. We clarify its output-delay quotient: the ten displayed complete-output residuals are distinct as a raw letter-to-letter kernel, but standard subsequential output-delay normalization gives a six-state quotient. We then study the operation of iterating the Berstel recoding until the greedy Zeckendorf normal form is reached. Binary cleanup has exact worst-case depth ⌊ L/2 ⌋ on trimmed binary words of length L , and on genuine addition inputs the sharp depth is max w∈Add 2 MSD (n) τ(w) = ⌈ n /2 ⌉; the maximum is already attained at significant length, without leading-zero padding. A standard local-propagation obstruction is recorded separately in an appendix.
In this paper, we introduce and study new matrix-domain sequence spaces generated by Perrin numbers. We construct an infinite lower triangular matrix 𝒫 = (𝒫˜ nk ) and define the associated sequence spaces ℓ 𝒫 ( 𝒫 ), ℓ ∞ ( 𝒫 ), c ( 𝒫 ), and c 0 ( 𝒫 ). We investigate several algebraic, topological, and geometric properties of these spaces. We further determine Schauder bases and study various inclusion relations among the Perrin sequence spaces. Moreover, we examine certain geometric properties including the approximation property, the Kadec–Klee property, the Hahn–Banach extension property, and rotundity. The α -, β -, and γ -duals of the newly introduced spaces are also computed. In addition, we characterize several classes of infinite matrix transformations involving Perrin sequence spaces and obtain necessary and sufficient conditions for matrix mappings of the forms ( X ( 𝒫 ), Y ) and ( X ( 𝒫 ), Y ( 𝒫 )), where X , Y ∈ { ℓ p , ℓ ∞ , c , c 0 }.
Catastrophic fault patterns are failure patterns resulting in the loss of connectivity of a processor array. We provide bijective correspondences between such patterns in one-dimensional processor arrays and lattice paths satisfying certain inertial criteria. In that way, we obtain new results for the asymptotic behavior of the enumerating sequences of catastrophic fault patterns.We also recover and reinterpret several known enumerative results and open possibilities for obtaining analogous results for more complex, but still essentially one-dimensional arrays.
Let Δ 2 ( x ) denote the error term in the asymptotic formula for the summatory function of d ( n ). Let Δ 1 ( x ; φ) denote the error term in the asymptotic formula of the Riesz mean. This paper devoted to the study of the hybrid moments of Δ 2 ( x ) and Δ 1 ( x ; φ).
Let 1 < c < (300s + 31)/(200s + 64) be fixed. Assume that N > 0 is a large enough number and ε > 0 is an arbitrarily small constant. This paper establish that the Diophantine inequality | p 1 c + … + p s c − N | < ε has a solution in prime numbers p 1 ,… ,p s , where s ≥ 7 and s is a natural number and p 1 can be expressed as p 1 = m 2 + n 2 + 1 for some integers m,n.
This paper establishes explicit evaluations of the 2 k -th power mean for generalized cubic Gauss sums. By exploiting analytic techniques and fundamental properties of classical Gauss sums, we derive closed-form expressions for these means. Furthermore, we develop a computationally efficient framework for analyzing higher-order moments of such sums.
Let Delta 2(x) denote the error term in the asymptotic formula for the summatory function of d(n). Let Delta 1(x; phi) denote the error term in the asymptotic formula of the Riesz mean. This paper devoted to the study of the hybrid moments of Delta 2(x) and Delta 1(x; phi).
In the current paper, we prove that the sums in the title can be reduced as combinations of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
Let u, v be strings over an alphabet Sigma with |Sigma| >= 2. Let L(u, v) be the Levenshtein distance and, when |u| = |v|, let H(u, v) be the Hamming distance. For equal-length strings, L(u, v) <= H(u, v). For u is an element of Sigma*, definehl(u): = max| v |=| u | (H(u, v)-L(u, v)).Ruth and Lladser [Discrete Math. 346 (2023) Paper No. 113310] showed that if the run-length of u is at most 2, then hl(u) = 0; conversely, hl(u) = 0 implies rho(u) <= 2. For p is an element of {2, 3}, we characterize all strings u with hl(u) = |u| - p. We also identify two types of strings u such that hl(u) = |u| - 4.
In this paper, we introduce the Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order alpha. By employing the M-calculus, based on the Mersenne numbers, we establish explicit series representations, addition theorems, difference equations and convolution identities.
This paper establishes that for k >= 4, all sufficiently large even integers n <= N, with at most O(N1/2 - & vartheta;(k) + epsilon) exceptions, admit representations as the sum of one square of a prime, four cubes of primes and one kth power of a prime, where the exponent & vartheta;(k) relies on k. This result sharpens the bound previously obtained by [J. J. Li, F. Xue and M. Zhang, Bull. Aust. Math. Soc. 107 (2023) 416-431].
Let p1, p2, & mldr;, p7 be prime numbers. In this paper, we first show that when k1 >= 49, any sufficiently large odd integer can be represented as the sum of a prime square, six prime cubes and k1 powers of 2. Furthermore, we prove that for k2 >= 73, every pair of sufficiently large odd integers can be expressed as a pair of equations involving a prime square, six prime cubes and k2 powers of 2.
This paper develops a comprehensive theory of r -generalized Fibonacci quaternion sequences with periodic coefficients, extending and unifying the classical theory of Fibonacci quaternion sequences. We establish explicit combinatorial formulas for both constant and periodic coefficients cases with particular emphasis on the role of periodicity in the sequence structure. Special attention is given to the case r = 2, leading to novel applications for Pell, h -Pell, and Pell-Lucas quaternion sequences. Our results generalize several existing theorems in the literature, while providing new insights into the combinatorial nature of r -generalized Fibonacci quaternion sequences.
Let lambda(1,) lambda(2), lambda(3), lambda(4), lambda(5 )be nonzero real numbers, neither all positive nor all negative and lambda(1)/lambda(2 )be an irrational number. Let V be a well-spaced sequence and delta > 0. For arbitrary epsilon > 0, we show that the quantity of v is an element of V with v <= N making the inequality |lambda(1)p(1)(2) + lambda(2)p(2)(2) + lambda(3)p(3)(4) + lambda(4)p(4)(4 )+ lambda(5)p(5)(4) - v| < v-delta unsolvable in primes p(1),p(2),p(3),p(4),p(5 )does not exceed O(N-6/7 (+ 2 delta + epsilon)).
Let β be a non-unit real algebraic integer greater than one and {a_n}_n ≥ 0 be a sequence satisfying a linear recurrence relation a_n+3=aa_n+2+ba_n+1+ca_n. Under certain conditions, we prove that the number of a_n which are palindromic concatenations of two repdigits in base β is finite.
Let k >= 7 be fixed, and v = (55k + 556)/(26k + 672),1 < d < c < v, 1 < mu < nu < k(1-d/c) and mu <= N-2/N-1(d/c) <= v. Suppose that N '(1), N '(2) > 0 are real numbers, and that N-1, N-2 are real numbers satisfying N-1 > N '(1) and N-2 > N '(2). Then we prove the system of two Diophantine inequalities |p(1)(c) + . . . + p(k)(c )- N-1| < N-1(-(1/c)(v-c)) log(195) N-1, |p(1)(d) + . . . + p(k)(d) - N-2 | < N(2)(-(1/d)(v-d) )log(195) N-2 has prime solutions p(1), . . . ,p(k).
In Chapter 3 of his second notebook, Ramanujan defined numbers a (n, k) such that a (2, 0) = 1 and for n >= 2, a (n + 1, k) = (n - 1) a (n, k - 1) + (2n - 1 - k) a (n, k), where a(n, k) = 0 when k< 0 or k>n - 2. These numbers are expressed in terms of Stirling numbers of the first kind and associated Stirling numbers of the second kind, and satisfy certain divisibility properties. In this paper, we obtain further properties for a(n, k), including a new characterization of prime numbers. We also derive several congruences for the numbers of Ramanujan modulo p(2), some of which lead to new conditions for a prime number to be a Wilson prime.
Let 1 < c < (300s + 31)/(200s + 64) be fixed. Assume that N > 0 is a large enough number and epsilon > 0 is an arbitrarily small constant. This paper establish that the Diophantine inequality |p(1)(c) + & mldr; + p(s)(c) - N | < epsilon has a solution in prime numbers p(1),& mldr;,p(s), where s >= 7 and s is a natural number and p(1) can be expressed as p(1) = m(2) + n(2) + 1 for some integers m,n.
This paper establishes explicit evaluations of the 2k-th power mean for generalized cubic Gauss sums. By exploiting analytic techniques and fundamental properties of classical Gauss sums, we derive closed-form expressions for these means. Furthermore, we develop a computationally efficient framework for analyzing higher-order moments of such sums.
