In the special theory of relativity, there is an unusual formula for addition of velocities. We will use this formula to generate Pythagorean triples. We define a Pythagorean triple in the usual manner as a triple of positive integers (a, b, c) such that a2 + b2 = c2. The numbers a and b are called legs, and c is called the hypotenuse. Later, we will allow b to take on the value of any integer. A primitive Pythagorean triple is a Pythagorean triple for which a, b and c are relatively prime.

The 2 × 2 identity matrix, $${I_2} = \left( \begin{gathered}{\rm{1 \,\,\,0}} \hfill \\{\rm{0 \,\,\,1}} \hfill \\ \end{gathered}\right)$$, has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples.

When children are first taught about multiplication of the natural numbers, it is usually presented as repeated addition. Later, learning about raising to an exponent is presented as repeated multiplication. Then the following analogy is obvious: addition is to multiplication as multiplication is to raising to an exponent. An interesting question is to ask what happens if we go in the other direction. That is, multiplication is to addition as addition is to what? In this paper, we will answer this question, and show that there are several possible operations that could be used to answer the question. Some of them will be presented in connection with algebraic semirings. A semiring is a set, together with two operations, semiring addition and semiring multiplication, satisfying certain properties. We will define several semirings, such that the semiring multiplication is ordinary addition. In each case, the semiring addition will then be an answer to the question in the title of this paper.

Let us define a Fibonacci-Lucas hyperbola as a hyperbola passing through an infinite number of points of the form (Fm, Ln), where the Fm are distinct Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21,…, where F0 = 0), and the Ln are distinct Lucas numbers (2, 1, 3, 4, 7, 11, 18, 29,…,. where L0 = 2). The simplest examples are 5x2 - y2 = 4, which contains the points (Fk, Lk) with odd subscripts, e.g. (1, 1), (2, 4), (5, 11), and 5x2 - y2 = -4, which contains the points with even subscripts, e.g. (0, 2), (1, 3), (3, 7); (see [1, 2]). These follow immediately from the identity (1)

$$L_n^2 - 5F_n^2 = 4{( - 1)^n}.$$

We define a Fibonacci fraction circle to be a circle passing through an infinite number of points whose coordinates are of the form $\left( {{{{F_k}} \over {{F_m}}},{{{F_n}} \over {{F_m}}}} \right)$ , where the F’s are Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …). For instance, the circle (1)

$${x^2} + {\left( {y + {1 \over 2}} \right)^2} = {5 \over 4}$$

$\left( {{1 \over 1},{0 \over 1}} \right),\,\left( {{1 \over 2},{1 \over 2}} \right),\,\left( {{1 \over 5},{3 \over 5}} \right),\,\left( {{1 \over {13}},{8 \over {13}}} \right),\left( {{1 \over {34}},{{21} \over {34}}} \right),\,...,$

Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.

Some favourite topics of both mathematics teachers and mathematics students are the golden ratio, Fibonacci numbers, groups, and Pythagorean triples. The material of this paper involves the first three, and there are some analogies with the fourth.