2026· Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös nominatae Sectio computatorica· 0 citations
Abstract
Let L be the class of all nonconstant polynomials f(x) == Ax2 + Bx + C ∈ Z[x], which are not of the forms Ax2, Bx. We
consider the directed graph Gf whose vertex set is P and p → q iff q | f(p), p ̸= q. Let Lf (p) be the set of simple directed paths starting at p and ℓf (p) = sup{|L| : L ∈ Lf (p)}, where |L| is the number of vertices in L. We prove supp∈P ℓf (p) = ∞. We study linear polynomials f(x) = Bx + C ∈ Z[x], BC ̸= 0, in connection with Cunningham chains. For f(x) = x 2+1, we explicitly construct a simple directed path with 130 vertices starting from 2, hence ℓx2+1(2) ≥ 130. It remains open whether
ℓx2+1(2) is finite or infinite. For these topics, we refer to works of C. Frayer [1], A.J. Pollington [6], C. Pomerance [7] and M. van RossumWijsmuller [8].
Let p ≥ 5 be prime, and let g(2n, p) denote the number of unordered representations
by positive integers
2n = h + k, h ≤ k, gcd(h, 6p) = gcd(k, 6p) = 1.
Every positive integer coprime to 6 belongs to exactly one of the families 6z + 1 and
6z + 5. In each family, divisibility by p excludes one explicit residue class of...
Andres Salazar· Communications on Number The...· 1 citation
Let G(V,E) be a graph, and let f: V(G) →{1,2,….|𝑉(𝐺)|} be a bijective vertex labeling. The induced edge labeling f*: E(G) → {0,1} is defined by assigning the label 1 to an edge uv whenever the labels f(u) and f(v) have same parity, and the label 0 whenever they have different parities. If the difference between the num...
S. N. Kumar, D. Bharathi· International Journal of Mat...· 0 citations
Let p be an odd prime, let q_0 := p^s with s>= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the class...
Let A = F_2[T]. The orthogonal multiplicity m(f) of a monic f in A of degree d>= 1 is the number of g in A with deg g<d, gcd(g,f) = 1, and all partial quotients of g/f of degree one. Blackburn (1998) asked how the proportion p(d) of monic polynomials of degree d with m(f)>0 behaves as d tends to infinity. We attach to...
