Pythagorean Triangles

So, you think you know everything about Pythagorean triangles? What if I told you that you could learn something new about them from this short book? And that all the results in this book are deduced using basic algebra and modular arithmetic?

Okay, most of you probably wouldn't learn anything new. Much of the material is common knowledge for most mathematicians. But, for the rest of you, here's what you could learn about. This book, by Wacław Sierpiński, was originally published in 1954 in Warsaw, with the English translation provided by Dr. Ambikeshwar Sharma in 1962. This re-release comes through Dover Publications.

A Pythagorean triangle is a right triangle whose side lengths are positive integers. And, of course, thanks to Pythagoras, we know that if the two side lengths are x and y, and the hypotenuse is z, then we have x² + y² = z². The two sides x and y are called legs or (as in this book) arms (whichever appendage you prefer). Some of the highlights:

Chapter 1: The author draws attention to primitive Pythagorean triangles (meaning no two of the integers x, y, and z have a common factor greater than one).

Chapter 2: With the help of some modular arithmetic, the standard description of Pythagorean triangles is given in Theorem 1, along with a variation in Theorem 2. The author finishes the chapter by noting that the radius of the circle inscribed in this Pythagorean triangle is (x + y – z)/2.

Chapter 3: A list of all of the Pythagorean triangles with side lengths less than 100 is given.

Chapter 4: This chapter has some of the most interesting results in the book, including: a classification of all Pythagorean triangles whose sides form an arithmetic progression; a classification of all Pythagorean triangles where the hypotenuse is one unit longer than one of the arms; and a classification of all Pythagorean triangles in which the two arms are consecutive integers. In fact, an algorithm is given to generate all of the latter triangles. We also get a one-to-one correspondence between Pythagorean triangles in which the two sides are consecutive integers and triangular numbers which are also squares (although there is a typo in the work here: in section 4.9, the author seems to be using u = z – x – 1, rather than u = z – x).

Chapter 5: We find that at least one of the arms of a Pythagorean triangle is a multiple of 3, while at least one of the three sides (arms or hypotenuse) must be a multiple of 5.

Chapter 6: We find which integers can appear as an arm or hypotenuse of a Pythagorean triangle. If n is greater than 2, then there is a Pythagorean triangle having n as an arm. For n to be a hypotenuse, it must have at least one prime factor congruent to 1 modulo 4 (no proof of this is given). We are told (also with no proof) that for any positive integer m, there exists a positive integer n such that there are m Pythagorean triangles with hypotenuse n, n + 1, n + 2, … , n + m – 1. We are also told that it is unknown whether there exists infinitely many Pythagorean triangles in which one arm and the hypotenuse are primes. The chapter closes with a summary of some asymptotic relations on the number of primitive Pythagorean triangles whose hypotenuses, perimeters, or areas are less than a given integer n.

Chapter 7: Constructive proofs are given to show that for any positive integer n, there exist at least n different Pythagorean triangles with the same arm and at least n different Pythagorean triangles with the same hypotenuse. The author mentions that it would be difficult to prove the corresponding results for primitive Pythagorean triangles (although, he says, the results are true as well).

Chapter 8: The author gives a way to construct n non-congruent Pythagorean triangles with the same perimeter.

Chapter 9: Similar to the previous chapter, we find a way to construct n non-congruent Pythagorean triangles with the same area. Also, a proof is given which shows that there are infinitely many right triangles with rational side lengths whose area is 6.

Chapter 9a: We find that the radius of the inscribed circle for a Pythagorean triangle and the radii of the three possible circumscribed circles are all integers. We are also given the exact number of primitive Pythagorean triangles with a given integer as the radius of the inscribed circle.

Chapter 10: We get several results involving the existence of Pythagorean triangles in which one or more sides, or the area, is a square or cube. This includes a (tenuous) connection with Fermat's Last Theorem, which at the time of this book's writing had only been verified for exponents less than 4003 (and other special cases).

Chapter 11: This chapter shows how to piece two Pythagorean triangles together to form a (not necessarily right) triangle with integral sides and integral area. Interestingly, not every triangle with this property can be formed by piecing together two Pythagorean triangles. However, any triangle with rational sides and area can be formed by piecing together two right triangles with rational sides. The chapter ends with a result of Kummer involving the intersection of diagonals of quadrilaterals.

Chapter 12: This is the most involved — mathematically speaking — of the chapters. It is dedicated to proving Fermat's assertion that the smallest Pythagorean triangle in which the hypotenuse and the sum of the arms are squares is the triangle (456548602761, 1061652293520, 4687298610289).

Chapter 13: This chapter gives the familiar correspondence between Pythagorean triangles and rational points on the unit circle.

Chapter 14: We get a description of all right triangles whose sides are given by (1/ x, 1/y, 1/z), where x, y, and z are integers.

Chapter 15: The final chapter moves into three dimensions, giving a classification of rectangular solids (cuboids) whose three side lengths and diagonal are all integers. (In 15.21, the author mentions that it is unknown whether there exists a cuboid for which the three side lengths, the three side diagonals, and the main diagonal are all integral. This problem, known as the brick problem or the perfect cuboid problem, remains open.)

All in all, this is an interesting collection of facts about Pythagorean triangles. While most of the mathematics consists of basic algebra and arithmetic, much of the arithmetic could be simplified by using congruence arguments. (For example, sections 2.1, 5.1, 5.3, 15.8.) Nonetheless, this is a fun book for a mostly general audience (at least, any audience that isn't scared of algebra).

Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at vestal@mwsc.edu.