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We study the number of ordered k-partitions of a multiset with equal sums, having elements $$\alpha _1,\ldots ,\alpha _n$$ α 1 , … , α n and multiplicities $$m_1,\ldots ,m_n$$ m 1 , … , m n . Denoting this number by $$S_k(\alpha _1, \ldots , \alpha _n; m_1, \ldots , m_n)$$ S k ( α 1 , … , α n ; m 1 , … , m n ) , we find the generating function, derive an integral formula, and illustrate the results by numerical examples. The special case involving the set $$\{1,\dots ,n\}$$ { 1 , ⋯ , n } presents particular interest and leads to the new integer sequences $$S_k(n)$$ S k ( n ) , $$Q_k(n)$$ Q k ( n ) , and $$R_k(n)$$ R k ( n ) , for which we provide explicit formulae and combinatorial interpretations. Conjectures in connection to some superelliptic Diophantine equations and an asymptotic formula are also discussed. The results extend previous work concerning 2- and 3-partitions of multisets.

The semiperimeter s, inradius r and circumradius R are the symmetric invariants of a triangle. In this short note, we complement previous work of Hirakawa and Matsumura by determining all pairs (up to similitude) consisting of a rational right angled triangle and a rational isosceles triangle having two corresponding symmetric invariants equal. In the proof of our main theorems, we determine the set of rational points on two hyperelliptic curves of genus 3. The first of these can be settled via a classical descent argument, whereas the second one is proved using ‘‘elliptic curve Chabauty’’.

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

We study the direct product of the automorphism groups of a finite number of groups. This is always isomorphic to a subgroup of the automorphism group of the direct product of the initial groups an...

We present the elliptic equation X3 + 2 = Y 2 as the rst in a sequence of Diophantine equations arising from some new results in the theory of partitions of multisets with equal sums. Two proofs for Theorem 2.3, showing that the only integer solutions to this equation are (-1; 1) and (-1

For two positive integers k,n≥2 we consider the equation xk≡1 (modn) and we let s > 1 be its smallest nontrivial integer solution. We prove that as n ranges over the positive integers, the ratio (s...

Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1,?, mn,?, and on a sequence of complex numbers z1,?, zn, ? of modulus one. We investigate some particular instances such as: extended cyclotomic, extended polygonal-type, and multinomial polynomials, for which we obtain formulae for the coefficients. Some novel related integer sequences are also derived.

Using the geometric dynamic of an iterative process (Theorem 2.1), we obtain refinements to some famous geometric inequalities in a triangle by constructing interpolating sequences.

Partitions play an important role in numerous combinatorial optimization problems. Here we introduce the number of ordered 3-partitions of a multiset M having equal sums denoted by S(m1,…, mn; α 1 ,…, α n ), for which we find the generating function and give a useful integral formula. Some recurrence formulae are then established and new integer sequences are added to OEIS, which are related to the number of solutions for the 3-signum equation.

Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.

In this chapter we present fundamental concepts and motivating examples of recurrent sequences, and show connections of recurrence relations to mathematical modeling, algebra, combinatorics, and analysis. There are numerous sources presenting the classical theory.

The sequence (xn)n≥0 = (xn(a, b;p, q))n≥0 defined by $$\displaystyle x_{n+2} = px_{n+1} + qx_n, \quad x_0 = a, x_1 = b, $$ with a, b, p, and q arbitrary complex numbers is called a Horadam sequence. For simplicity, we shall denote (xn(a, b;p, q))n≥0 by (xn)n≥0 hereafter.

This chapter presents key properties of the complex linear recurrent sequences of arbitrary order, including general term formulae, periodicity, or the asymptotic behavior in some particular cases. We also give results for systems of linear recurrent sequences.

In this chapter we present basic results and examples of first-order and second-order linear recurrent sequences with arbitrary coefficients, some classical sequences, polynomials, and linear fractional transformations.

In this chapter we start by presenting some key results concerning the number of ordered k-partitions of multisets with equal sums. For these we give generating functions, recurrences and numerical examples. The coefficients arising from these formulae are then linked to certain elliptic and superelliptic Diophantine equations, which are investigated using some methods from Algebraic Geometry and Number Theory, as well as specialized software tools and algorithms. In this process we are able to solve some recent open problems concerning the number of solutions for certain Diophantine equations and to formulate new conjectures.

This material is a systematic presentation of the exponent of a group with properties, examples, and explicit computations. Some of the results and connexions we present are original or they are presented in a new form. The last section explores general results about the exponent of certain finite groups of matrices. We will explicitly compute the exponent for \( \operatorname {\mathrm {SL}}_2(\mathbb Z/p^n \mathbb Z)\) and \( \operatorname {\mathrm {GL}}_2(\mathbb Z/p^n \mathbb Z)\), when p ∈{2, 3} and n is a positive integer.

In this chapter we present a thematic collection of 123 olympiad training problems, involving recurrences in various contexts. We have chosen 123 as it represents the 10th Lucas number L10. The problems relate to topics such as linear recurrence sequences of first, second, and higher orders, classical sequences, homographic recurrent sequences, systems of recurrent sequences, and combinatorics.

In this chapter we present detailed solutions (sometimes two or three) for the proposed problems in Chapter 7.

The classical second-order recurrent sequences discussed in Chapter 2 have many interesting properties. In Section 3.1 we present arithmetic properties of the Fibonacci and Lucas sequences first found in [23]. Section 3.2 studies properties of Pell and Pell–Lucas sequences. We also discuss about certain notions of Fibonacci, Lucas, Pell, or Pell–Lucas primality. These results have been extended to generalized Lucas and Pell–Lucas sequences in our paper (Andrica and Bagdasar, Mediterr. J. Math. 2021, to appear). Section 3.3 presents factorizations of the terms of classical sequences as products of trigonometric expressions. These are derived from the complex factorizations of the general polynomials Un and Vn ( 2.60), and some involve the resultant of polynomials.

We prove two properties regarding the Fibonacci and Lucas Sequences modulo a prime and use these to generalize the well-known property p ∣ F p − p 5 . We then discuss these results in the context of primality testing.

We prove that sums of the form with f(X), g(X) ∈ ℤ[X] can be explicitly computed whenever f and g are subject to some certain conditions which are defined in the paper.

While units in any unital ring are strongly clean by denition, which units are uniquely clean, is a far from being simple question, even in particular rings. In this paper, the question is solved for 2 2 integral matrices. It turns out that uniquely clean invertible matrices are scarce: only the matrices similar to [1 0; 0 -1]. The study is splitted into three cases: the elliptic, the parabolic and the hyperbolic cases, according to the discriminant of their characteristic polynomial. In the rst two cases, units are not uniquely clean.

We present some new methods for constructing convex functions. One of the methods is based on the composition of a convex function of several variables which is separately monotone with convex and concave functions. Using several well-known results on the composition of convex and quasi-convex functions we build new convex, quasi-convex, concave, and quasi-concave functions. The third section is dedicated to the study of convexity property of symmetric Archimedean functions. In the fourth section the asymmetric Archimedean function is considered. A classical example of such a function is the Bellman function. The fifth section is dedicated to the study of convexity/concavity of symmetric polynomials. In the sixth section a new proof of Chandler–Davis theorem is given. Starting from symmetric convex functions defined on finite dimensional spaces we build several convex functions of hermitian matrices. The seventh section is dedicated to a generalization of Muirhead’s theorem and to some applications of it. The last section is dedicated to the construction of convex functions based on Taylor remainder series.

We present some new aspects involving strong convexity, the pointwise and uniform convergence on compact sets of sequences of convex functions, circular symmetric inequalities and bistochastic matrices with examples and applications, the convexity properties of the multivariate monomial, and Schur convexity.

In this paper we introduce several natural sequences related to polynomials of degree s having coefficients in {1,2, …,n} (n ∈ N) which factor completely over the integers. These sequences can be seen as generalizations of A006218. We provide precise methods for calculating the terms and investigate the asymptotic behavior of these sequences for s ∈ {1, 2, 3}.

The interplay between integer sequences and partitions has led to numerous interesting results, with implications in generating functions, integral formulae, or combinatorics. An illustrative example is the number of solutions at level n to the signum equation. Denoted by S(n), this represents the number of ways of choosing + and − such that ±1±2±3±···±n = 0 (see A063865 in OEIS). The Andrica-Tomescu conjecture regarding the asymptotic behaviour of S(n) was solved affirmatively in 2013, and new conjectures were formulated since then. In this paper we present recurrence formulae, generating functions and integral formulae for the number of ordered 2-partitions of the multiset M having equal sums. Certain related integer sequences not currently indexed in the OEIS are then presented. Finally, we formulate conjectures regarding the unimodality, distribution and asymptotic behaviour of these sequences.