A STUDY ON PYTHAGOREAN TRIPLES

The Pythagorean numbers play a significant role in the theory of higher arithmetic as they come in the majority of indeterminate problem. For the discovery of the law of the three squares (Pythagorean equation), really, one should be indebted to the Pythagorean who were the first Greeks with great intellectual perception. One may notice to his surprise that the Egyptians, the Chinese, the Babylonians and the Indians knew some knowledge of the property of right angled Pythagorean triangles or Pythagorean numbers .Since there is a 1−1 correspondence between Pythagorean numbers and Pythagorean triangles, we shall use them interchangeably. The only geometrical theorem with which the ancient Chinese were acquainted is that the area of the square described on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares described on the sides. A Pythagorean triangle is a right triangle whose sides are integral lengths

Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.Comment: 11 pages, 1 figur

Levels of Distribution and the Affine Sieve

This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve theory which have received attention recently: (1) the spectacular work of Yitang Zhang, under the title "Level of Distribution," and (2) the so-called "Affine Sieve," introduced by Bourgain-Gamburd-Sarnak.Comment: 34 pages, 2 figure

Incremental and Transitive Discrete Rotations

A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from \ZZ[i] to \ZZ[i], whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotate d copies of an image for angles in-between 0 and a destination angle. The di scretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm whic h computes incrementally a discretized rotation. The suggested method uses o nly integer arithmetic and does not compute any sine nor any cosine. More pr ecisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be th e key ingredient that will make the resulting procedure optimally fast and e xact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of defin itions for discrete geometry

How many primitive Pythagorean triples in arithmetic progression

Everyone knows that (3, 4, 5) is a Pythagorean triple (‘PT’); for, the numbers satisfy the Pythagorean relation 3^2 +4^2 =5^2. Indeed, it is a Primitive Pythagorean triple (‘PPT’) since the integers in the triple are coprime. (See the Problem Corner for definitions of unfamiliar terms.) But this triple has a further property: its entries are in arithmetic progression for, 3, 4, 5 forms a three-term AP with common difference 1. Naturally, our curiosity is alerted at this point