Search

The main results characterize the equations (2.1) and (2.10) whose solutions are linear combinations with rational coefficients of at most two terms of classical Fibonacci and Lucas sequences.

Using the system (1.4) in Theorem 1.1, we present in Section 3 a new way to derive the Rodrigues formula for the exponential map of the Lorentz group SO(3, 1).

In Theorem 1 we present, in the case when the eigenvalues of the matrix are pairwise distinct, a direct way to determine the general Rodrigues coefficients of a matrix function for the general linear group \(\mathbf{GL}(n, \mathbb{R})\) by reducing the Rodrigues problem to the system (7). Then, Theorem 2 gives the explicit formulas in terms of the fundamental symmetric polynomials of the eigenvalues of the matrix. Our formulas permit to consider also the degenerated cases (i.e., the situations when there are multiplicities of the eigenvalues) and to obtain nice determinant formulas. In the cases n = 2, 3, 4, the computations are effectively given, and the formulas are presented in closed form. The method is illustrated for the exponential map and the Cayley transform of the special orthogonal group SO(n), when n = 2, 3, 4.

Euler, after a cursory reading of Wallis’s Opera Mathematica, mistakenly attributed the first serious study of nontrivial solutions to equations of the form \(x^{2} - Dy^{2} = 1\), where x ≠ 1 and y ≠ 0, to John Pell. However, there is no evidence that Pell, who taught at the University of Amsterdam, had ever considered solving such equations. They should be probably called Fermat’s equations, since it was Fermat who first investigated properties of nontrivial solutions of such equations.

This chapter gives the general theory and useful algorithms to find positive integer solutions (x, y) to general Pell’s equation (4.1.1), where D is a nonsquare positive integer, and N a nonzero integer.

An interesting problem concerning the Pell’s equation \(u^{2} - Dv^{2} = 1\) is to study when the second component of a solution (u, v) is a perfect square.

This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis

In order to motivate the study of quadratic type equations, in this chapter we present several problems from various mathematical disciplines leading to such equations. The diversity of the arguments to follow underlines the importance of this subject.

In 1900, David Hilbert asked for an algorithm to decide whether a given Diophantine equation is solvable or not and put this problem tenth in his famous list of 23.

The main goal of this chapter is to lay out basic concepts needed in our study in Diophantine Analysis. The first section contains fundamental results pertaining to continued fractions, some without proofs. The Theory of Continued Fractions is not new but it plays a growing role in contemporary mathematics.

The paper is an addendum to D. Andrica and L. Funar, 'On smooth maps with finitely many critical points', J. London Math. Soc. (2) 69 (2004) 783-800.

We go through a series of results related to the k-signum equation \(\pm 1^k\pm 2^k\pm\cdots\pm n^k=0\). We are investigating the number S k (n) of possible writings and the asymptotic behavior of these numbers, as k is fixed and \(n\to \infty\). The results are presented in connections with the Erdos–Suranyi sequences. Analytic methods and algebraic ones are employed in order to predict the asymptotic behavior in general and to study in detail various situations for small values of k. Some simplifications and further ramifications are discussed in the end about the recent proof of Andrica–Tomescu conjecture.

The second chapter is devoted to the study of the trigonometric form of complex numbers and it contains two sections dealing with the following aspects: polar representation of complex numbers, and the nth roots of unity. Suggestive examples and problems of various level of difficulties are included.

The last chapter presents answers, hints or complete solutions to all suggested problems and it contains two sections. Section 6.1 is devoted to the routine problems and section 6.2 to the olympiad-caliber problems. For many problems alternative solutions are provided.

The fourth chapter is dealing with some deep aspects of the plane Euclidean geometry illustrated by the using of complex coordinates, and it contains a dense material organized into 13 sections. Sections 4.1 and 4.2 present the real product and the complex product, new operations with complex numbers systematically used in the next parts of the book. The other sections of this chapter are dealing with the following aspects: area of a convex polygon in complex coordinates, important points in triangle geometry defined by intersecting Cevians, the nine-point circle of Euler, some important distances in triangle computed in terms of fundamental invariants, the using of barycentric coordinates, orthopolar triangles, area of the antipedal triangle, Lagrange’s theorem and some of its applications. A special attention is paid to the presentation of the basic geometric transformations of the complex plane. An important part of the chapter is given by applications, examples and problems, that complete in the best way the theoretical aspects.

This chapter is dealing with the first connections between complex numbers and geometry and it is organized into six sections containing a rich material involving the following aspects: simple geometric notions and properties in the language of complex numbers, conditions for collinearity, orthogonality and concyclicity in terms of complex numbers, similar triangles, equilateral triangles, elements of analytic geometry in the complex plane, the study of circles. The theoretical material is illustrated by suggestive examples and problems of various level of difficulty.

This chapter is a special feature of the book and it is an outstanding selection of genuine olympiad and other important mathematical contest problems solved using the methods and techniques already presented. The problems are organized into nine sections as follows : problems involving moduli and conjugates, algebraic equations and polynomials, connections between algebraic identities and geometric properties, geometric problems, trigonometric problems, problems involving the nth roots of unity, problems involving polygons, complex numbers and combinatorics, miscellaneous problems.

The first chapter is devoted to the presentation of the basic concepts and results involving the algebraic form of complex numbers. The material is organized into two sections realizing a first connection to the plane geometry : algebraic representation of complex numbers, and geometric interpretation of the algebraic operations. Suggestive examples and problems are included.

We deal with the minimum number of critical points of circular functions with respect to two different classes of functions. The first one is the whole class of smooth circular functions and, in this case, the minimum number is the so called circular \(\varphi\) -category of the involved manifold. The second class consists of all smooth circular Morse functions, and the minimum number is the so called circular Morse–Smale characteristic of the manifold. The investigations we perform here for the two circular concepts are being studied in relation with their real counterparts. In this respect, we first evaluate the circular \(\varphi\)-category of several particular manifolds. In Sect. 5, of more survey flavor, we deal with the computation of the circular Morse–Smale characteristic of closed surfaces. Section 6 provides an upper bound for the Morse–Smale characteristic in terms of a new characteristic derived from the family of circular Morse functions having both a critical point of index 0 and a critical point of index n. The minimum number of critical points for real or circle valued Morse functions on a closed orientable surface is the minimum characteristic number of suitable embeddings of the surface in \(\mathbb{R}^{3}\) with respect to some involutive distributions. In the last section we obtain a lower and an upper bound for the minimum characteristic number of the embedded closed surfaces in the first Heisenberg group with respect to its noninvolutive horizontal distribution.

The main result is given in Theorem 2 and it shows that the inequality γ ( M) kγ (M)−4(k−1) contained in Theorem 1 holds with equality for any closed smooth surface M2 (orientable or not), where π : M → M is a k -covering of M and γ (M) is the Morse-Smale characteristic of M . Mathematics subject classification (2000): 57R70, 58E05, 26D99.

In the first part of this survey we review some special cases of \({\varphi }_{\mathcal{F}}\)-category of a pair (M, N) of manifolds such as φ-category, Morse-Smale characteristic, and Morse-Smale characteristic for circular functions. Section 2 presents examples of pairs with finite φ, and Sect. 3 provides lower estimates for the size of the critical sets in terms of topological dimension. We employ the cardinality when the manifolds admit maps with finitely many critical points and the topological dimension when no such maps exist.

The main purpose of this paper is to describe the isometry groups \(\mathit{Iso}_{d_{p}}(\mathbb{R}^{n})\) for p≥1, p≠2, and p=∞, where the metric d p is given by (4.2). A corollary of the main result contained in Theorem 4.1 and Theorem 4.2 is that in case p≠2 all these groups are isomorphic and, consequently, they are independent of p. In the last section, the isometry dimension of a finite group with respect to a given metric on the space ℝ n is introduced.

1. Solve the following equation in integers x, y : $$x^{2} + 6xy + 8y^{2} + 3x + 6y = 2.$$ Solution. Write the equation in the form $$(x + 2y)(x + 4y) + 3(x + 2y) = 2\quad{\rm or}\quad(x +2y)(x +4y + 3) = 2.$$ We obtain the solutions (0,−1), (3,−2), (3,−1), (6,−2).

A field is a set k equipped with two commutative binary operations, addition and multiplication, such that (k, +) is an abelian group under addition; every nonzero element of k has a multiplicative inverse, and (k*, ·) is an abelian group under multiplication, where k* = k \ {0k}; 0k ≠ 1k; the distributive law holds: (a + b)c = ac + bc for all a, b, c Є k.

In 1909, A. Thue proved the following important theorem: Let f = a n z n +a n -1zn-1+…+a1z+a0 be an irreducible polynomial of degree ≥ 3 with integral coefficients. Consider the corresponding homogeneous polynomial $$F(x,y)=y^nf\left(\frac{x}{y}\right)=a_nx^n+a_{n-1}x^{n-1}y+\cdots+a_1xy^{n-1}+a_0y^n.$$

1. Find all positive integers n such that \(\frac{{n}({n+1})}{3}\)is a perfect square. (Dorin Andrica) Soloution. Let \(\frac{{n}({n+1})}{3} = y^2,\) which is equivalent to $${(2n + 1)}^2 - 12y^2 = 1.$$

Solution. Working modulo 3, we have y ≡ 1 (mod 3); hence y = 1 + 3s, s ∈ ℤ. The equation becomes 6x − 15z = −9 − 30s, or equivalently, 2x − 5z = −3 − 10s. Passing to modulo 2 yields z ≡ 1 (mod 2), i.e., z = 1 + 2t, t ∈ ℤ and x = 1 -5s + 5t. Hence the solutions are $$(x,y,z)=(1 -5s + 5t,1 + 3s,1 +2t),\quad s,t \in \mathbb{Z}.$$

An equation of the form $$a_1x_1 + \cdots + a_n x_n = c,$$ (2.1.1) where a1, a2, …, a n , b are fixed integers, is called a linear Diophantine equation. We assume that n ≥ 1 and that coefficients a1, …, a n are all different from zero.

Given the equation f(x1, x2, …, xn) = 0, we write it in the equivalent form $$f_1(x_1, x_2, \ldots , x_n)f_2(x_1, x_2, \ldots , x_n) \cdots f_k(x_1, x_2, \ldots, x_n) = a,$$

1.Solve the equation $$x^2 + 4 = y^n,$$ where n is an integer greater than 1. Solution. For n = 2, the only solutions are (0, 2) and (0, –2). For n = 3, we have seen in Example 4 that the solutions are (2, 2), (–2, 2), (11, 5), and (–11,5). Lef now n ≥ 4. Clearly, for n even, the equation is not solvable, since no other squares differ by 4. For n odd, we may assume without loss of generality that n is a prime p ≥ 5. Indeed, if n = q k , where q is an odd prime, we obtain an equation of the same type: x2 + 4 = (y k ) q .