Journal ArticleDOI
89.29 An extension of the fundamental theorem on rightangled triangles
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In this paper, it was shown that a given integer n can play the role of a specified side in either 0 or 2k−1 different PPTs, where k is the number of distinct prime factors of n.Abstract:
{3, 4, 5} is perhaps the most famous Pythagorean Triple with interest in such triples dating back many thousands of years to the ancient people of Mesopotamia. In this article, we shall consider such triples, with the restriction that the elements of these triples must not have any common factors they are Primitive Pythagorean Triples (PPTs). In particular, we shall consider the question of how many PPTs a given integer can be a member of. The answer to this simple question is, surprisingly, that a given integer n can play the role of a specified side in either 0 or 2k−1 different PPTs, where k is the number of distinct prime factors of n. Our result is a generalisation of what Fermat grandly called the Fundamental Theorem on right-angled triangles ([2], chapter 5), which states that:read more
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The higher arithmetic (5th ed.), by H. Davenport. Pp 189. £12 hard covers, £5 paperback. 1982. ISBN 0 521 24422 6/28678 6 (Cambridge University Press)
Journal Article
Visualizations and intuitive reasoning in mathematics
TL;DR: In this article, the role of visualizations and intuitive thinking in mathematics has been considered and historical and didactical examples have been used as well as smaller empirical studies at upper secondary school level and university level.
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Number Theory and the Queen of Mathematics
TL;DR: In this paper, Campbell and Zazkis explored the relationship between geometry and number theory and found that geometry played an important role in the early development of number theory in ancient Mesopotamia.
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Book
An Introduction to the Theory of Numbers
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Book
An introduction to the theory of numbers
Ivan Niven,H. S. Zuckerman +1 more
TL;DR: Divisibility congruence quadratic reciprocity and Quadratic forms some functions of number theory some diophantine equations Farey fractions and irrational numbers simple continued fractions primes and multiplicative number theory algebraic numbers the partition function the density of sequences of integers.
Book
Combinatorics: Topics, Techniques, Algorithms
TL;DR: In this article, the authors present a textbook aimed at second-year undergraduates to beginning graduates, which stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that constructive or algorithmic proof is more valuable than an existence proof.