In this note we consider the title Diophantine equation from both a theoretical as well as experimental point of view. In particular, we prove that for k = 4 , 6 k=4, 6 and each choice of the signs our equation has infinitely many coprime positive integer solutions ( x , y , a , b ) (x, y, a, b) such that no partial sum in the expression x 3 ± y 3 − ( a k ± b k ) x^3 \pm y^3-(a^k \pm b^k) vanishes. The same is true for each k ≢ 0 ( mod 4 ) k\not \equiv 0\pmod {4} and the equation x 3 ± y 3 = a k − b k x^3\pm y^3=a^k-b^k . For k = 5 , 7 k=5, 7 and all choices of the signs we computed all coprime positive integer solutions ( x , y , a , b ) (x, y, a, b) of x 3 ± y 3 = a k + b k x^3\pm y^3=a^k+b^k satisfying the condition b > a ≤ 50000 b>a\leq 50000 .

A Hilbert cube of dimension d is the set of integers H(a_0; a_1, …, a_d)=a_0+{0, a_1}+⋯+{0, a_d}={a_0+∑_i=1^dε_ia_i: ε_i∈{0,1}}. Brown, Erdős and Freedman asked whether the maximal dimension of a Hilbert cube in the set S={n^2: n∈ℕ} of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if H(a_0; a_1, …, a_d)⊂S∩ [0, N], then d≤ 7 loglog N for all sufficiently large values of N. Here we prove that there exist at least ≫ N^1/8 Hilbert cubes H(a_0; a_1, a_2, a_3) with a_0, a_1, a_2, a_3∈ [0,N] in the set of squares. Moreover, we prove that for each i, j∈{0, 1, 2, 3} with i<j, the set {a_i/a_j: H(a_0; a_1, a_2, a_3)⊂ S} is dense in the set of positive real numbers (in the Euclidean topology).

In this note we consider the title Diophantine equation from both a theoretical as well as experimental point of view. In particular, we prove that for k = 4,6 and each choice of the signs our equation has infinitely many coprime positive integer solutions (x, y, a, b) such that no partial sum in the expression x(3) +/- y(3)-(a(k) +/- b(k)) vanishes. The same is true for each k (sic) 0 (mod 4) and the equation x(3) +/- y(3) = a(k)-b(k). For k = 5,7 and all choices of the signs we computed all coprime positive integer solutions (x, y, a, b) of x(3) +/- y(3) = a(k) + b(k) satisfying the condition b < a <= 50000.

Let s_𝐰 be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights 𝐰=(w_j)_j≥ 0. We investigate Hankel determinants ℋ_𝐰(n) = [s_𝐰(i+j)]_0≤ i,j<n and derive a general recursion that allows us to effectively compute ℋ_𝐰(n) for all n. Applying it to the ordinary binary sum-of-digits, that is, w_j=1, we express ℋ_𝐰(n) in a closed form for several sequences of indices, including the remarkably simple ℋ_𝐰(⌈ 2^k+2/3⌉)= (-1)^(k+2)(k+3)/2(k+1). This yields an infinite family of explicit evaluations, giving a partial solution to a problem posed by Allouche and Shallit. Moreover, we closely study the specialization w_j=t^j, where the determinants become polynomials in t, and investigate their vanishing. For t=2ζ, where ζ is a root of unity, we show that the determinants vanish on a large structured set of indices, while the complementary is sparse but infinite. In addition to ℋ_𝐰(n), we consider Hankel determinants associated with the first difference of s_𝐰, obtaining an explicit product formula. This generalizes the results by Fokkink, Kraaikamp, and Shallit concerning Hankel determinants for the period-doubling sequence.

We introduce a new family of number sequences (f(n))n∈N, governed by the recurrence relationf(n)=af(n−un−1)+bf(n−un−2), where u=(un)n∈N is a sequence with values 0,1. Our study focuses on the properties of the sequence of quotients h(n)=f(n+1)/f(n) and its set of values V(f)={h(n):n∈N} for various u. We give a sufficient condition for finiteness of V(f) and automaticity of (h(n))n∈N, which holds in particular when u is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software Walnut. On the other hand, we prove that the set V(f) is infinite for other special binary sequences u, and obtain a trichotomy in its topological type when u is eventually periodic.

Let C_n be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each k≥ 3 , there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of x∈ℕ_+ such that C_xC_x+1 is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.

Let A be a subset of positive integers. By A-partition of n, we understand the representation of n as a sum of elements from the set A. For given i, n∈ℕ , by c_A(i,n) , we denote the number of A-partitions of n with exactly i parts. In the paper, we obtain several result concerning sign behaviour of the sequence S_A,k(n)=∑ _i=0^n(-1)^ii^kc_A(i,n) , where k∈ℕ is fixed. In particular, we prove that for a broad class 𝒜 of subsets of ℕ_+ , we have that for each A∈𝒜 , we have (-1)^nS_A,k(n)≥ 0 for each n, k∈ℕ .

Let a, Q∈Q be given and consider the set 𝒢(a, Q)={aQ^i: i∈N} of terms of geometric progression with 0th term equal to a and the quotient Q. Let f∈Q(x, y) and 𝒱_f be the set of finite values of f. It is known that for any quadratic form f, there are infinitely many pairs a, Q such that 𝒢(a, Q)⊂𝒱_f . In this note we prove that the same statement is true for reducible cubic forms. The main tool in the proof is the density of rational points on the quartic surface z^3=(x^2-u)(y^2-u) , where u∈Q∖{0} .

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function F of at most exponential growth satisfying the condition F(ℕ)⊂ℝ_+ , we have F(a)F(b)>F(a+b) for sufficiently large positive integers a, b. Moreover, we show that if the sequence (F(n))_n≥ n_0 is log-concave and lim sup _n→ +∞F(n+n_0)/F(n)

In this paper, we investigate the set S(n) of positive integer solutions of the title Diophantine equation. In particular, for a given n we prove boundedness of the number of solutions, give precise upper bound on the common value of sigma(2)(X-n) and sigma(n)(X-n) together with the biggest value of the variable x(n) appearing in the solution. Moreover, we enumerate all solutions for n <= 16 and discuss the set of values of x(n)/x(n-1) over elements of S(n).

Let a,Q∈Q be given and consider the set G(a,Q)={aQi:i∈N} of terms of geometric progression with 0th term equal to a and the quotient Q. Let f∈Q(x,y) and Vf be the set of finite values of f. We consider the problem of existence of a,Q∈Q such that G(a,Q)⊂Vf. In the first part of the paper we describe certain classes of rational functions for which our problem has a positive solution. In the second, experimental, part of the paper we study the stated problem for the rational function f(x,y)=(y2−x3)/x. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures.

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for k=4, 6 and each choice of the signs our equation has infinitely many co-prime positive integer solutions. For k=5, 7 and all choices of the signs we computed all co-prime positive integer solutions (x, y, a, b) satisfying the condition max{a, b}≤ 50000.

In this paper, we investigate the set [Formula: see text] of positive integer solutions of the title Diophantine equation. In particular, for a given [Formula: see text] we prove boundedness of the number of solutions, give precise upper bound on the common value of [Formula: see text] and [Formula: see text] together with the biggest value of the variable [Formula: see text] appearing in the solution. Moreover, we enumerate all solutions for [Formula: see text] and discuss the set of values of [Formula: see text] over elements of [Formula: see text].

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over Q \mathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B = ( B 0 , B 1 , B 2 , B 3 ) B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer b b (depending on B B ), the equation B 0 X 0 5 + B 1 X 1 5 + B 2 X 2 5 + B 3 X 3 5 = b B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b has infinitely many positive integer solutions.

In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type.

Let D-n((v)) be the nth generalized derangement number that is a generalization of the classic derangement number D-n = D-n((0)) . In this note, we investigate the set S-v of those integers n for which D(v) n is not a sum of three squares. We characterize the set S-0 and the set S(v )for odd values of v. We prove that in these cases the set S-v has natural density and compute its value. In particular, the natural density of S(0 )is equal to 1/24.

Let m be a positive integer and b_m(n) be the number of partitions of a non-negative integer n with parts being powers of 2, where each part can take m colors. We show that if m=2^k-1 , then the natural density of n such that b_m(n) cannot be represented as a sum of three squares exists, and equals 1/12 for k=1, 2 and 1/6 for k≥ 3 . In particular, for m=1 the equation b_1(n)=x^2+y^2+z^2 has a solution in integers if and only if n is not of the form 2^2k+2(8s+2t_s+3)+i for i=0, 1 and k , s are non-negative integers, and where t_n is the n th term in the Prouhet–Thue–Morse sequence. A similar characterization is obtained for the solutions in n of the equation b_2^k-1(n)=x^2+y^2+z^2 .

In this note we investigate the set S(n) of positive integer solutions of the title Diophantine equation. In particular, for a given n we prove boundedness of the number of solutions, give precise upper bound on the common value of σ_2(X_n) and σ_n(X_n) together with the biggest value of the variable x_n appearing in the solution. Moreover, we enumerate all solutions for n≤ 16 and discuss the set of values of x_n/x_n-1 over elements of S(n).

Let , where is the sum of binary digits function. The sequence is the well-known Prouhet–Thue–Morse sequence. In this note we study the sequence , where and for we define recursively as follows: . We prove several results concerning arithmetic properties of the sequence . In particular, we prove non-vanishing of for , automaticity of the sequence for each m, and other results.

We are interested in solving the congruences f3+g3+1≡0(modfg) and f4−4g2+4≡0(modfg) in polynomials f,g with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively. Our approach is based on Gröbner basis techniques and we do numerical experiences based on it. We mainly deal with the case degf≤2 and prove that there are no new families of cyclic cubic nor cyclic quartic fields. In case of cyclic quartic fields we obtained a new polynomial that was not discovered by Balady and Washington [2], it is given by t4+7890798742t3−37333446t2+38618t+1.