Titu Andreescu

An Introduction to Diophantine Equations

In this article we will only touch on a few tiny parts of the fiel d of linear Diophantine equations. Some of the tools introduced, however, will be useful in many other parts of the subject.

Algebra and Analysis

1. If the inequalities

Number Theory and Combinatorics

a - {{b}^{2}} > \frac{1}{4},{\text{ }}b - {{c}^{2}} > \frac{1}{4},{\text{ }}c - {{d}^{2}} > \frac{1}{4},{\text{ }}d - {{a}^{2}} > \frac{1}{4}

The Splitting Method and Double Sequences

hold simultaneously, then by adding them we obtain a+b+c+d−(a2+b2+c2+) > 1.

Some Applications of the Hamilton-Cayley Theorem

In this chapter we study real sequences, a special class of functions whose domain is the set N of natural numbers and range a set of real numbers. 1.1 Main Definitions and Basic Results Hypotheses non fingo. [“I frame no hypotheses.”] Sir Isaac Newton (1642–1727) Sequences describe wide classes of discrete processes arising in various applications. The theory of sequences is also viewed as a preliminary step in the attempt to model continuous phenomena in nature. Since ancient times, mathematicians have realized that it is difficult to reconcile the discrete with the continuous. We understand counting 1, 2, 3, . . . up to arbitrarily large numbers, but do we also understand moving from 0 to 1 through the continuum of points between them? Around 450 essential way. As he put it in his paradox of dichotomy: course) before it arrives at the end. Aristotle, Physics, Book VI, Ch. 9 A sequence of real numbers is a function f : N→R (or f : N∗→R). We usually write an (or bn, xn, etc.) instead of f (n). If (an)n≥1 is a sequence of real numbers and if n1 < n2 < · · · < nk < · · · is an increasing sequence of positive integers, then the sequence (ank)k≥1 is called a subsequence of (an)n≥1. n n≥1 is said to be nondecreasing (resp., increasing) if an ≤ an+1 (resp., an < an+1), for all n ≥ 1. The sequence (an)n≥1 is called “>”) instead of “≤” (resp., “<”).

The Extreme Value Theorem

Problem 3.4 Prove that the sum of any n entries of the table

The Number e

\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & \frac{1}{2} & \frac{1}{3} & \ldots & \frac{1}{n}\\[4pt]\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \ldots & \frac{1}{n+1}\\[4pt]\vdots & & & & \\[4pt]\frac{1}{n} & \frac{1}{n+1} & \frac{1}{n+2} & \ldots & \frac{1}{2n-1}\end{array}