Pythagorean Triples Formula, Definition, List, and Examples

Pythagorean Triples are sets of three positive integers (a, b, c) that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). This relationship is represented by the equation a ² + b ² = c ² . In this context, 'a' typically represents the length of the side opposite the right angle (perpendicular), 'b' represents the length of the side adjacent to the right angle (base), and 'c' represents the length of the hypotenuse.

The most well-known and smallest Pythagorean triple is (3, 4, 5), where 3 ² + 4 ² = 5 ² . This triple is a fundamental example of how the Pythagorean theorem works.

Pythagoras, the ancient Greek mathematician who lived around 570 BC, is famous for discovering this theorem. He had a keen interest in mathematics, science, and philosophy, and the Pythagorean theorem is one of his significant contributions to mathematics. The theorem describes a fundamental property of right-angled triangles, where the longest side (the hypotenuse) is related to the other two sides by this formula.

In a right-angled triangle, 'c' represents the length of the hypotenuse, which is the side opposite the right angle. 'a' represents the length of the side opposite the right angle, which is perpendicular, and 'b' represents the length of the side adjacent to the right angle, known as the base.

In this article, we will explore Pythagorean triples in detail, including their formula, lists of triples, methods to find them, examples, and proofs of the Pythagorean theorem.

What are Pythagorean Triples?

Pythagorean Triples, often referred to as Pythagorean Triplets, are sets of three positive integers (a, b, and c) that satisfy the Pythagorean theorem: a ² + b ² = c ² . In a right-angled triangle, this theorem establishes a fundamental relationship between the lengths of its sides.

Here's a brief introduction to the Pythagoras theorem and how it works:

Pythagoras Theorem:

Pythagoras, an ancient Greek mathematician, is credited with discovering this theorem. It's named after him because he is known for proving it. The theorem states that in a right-angled triangle:

The square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, it can be expressed as a ² + b ² = c ² , where 'a' and 'b' are the lengths of the two shorter sides (the perpendicular and the base), and 'c' is the length of the hypotenuse.

For example, consider the Pythagorean Triple (3, 4, 5):

Here, a = 3, b = 4, and c = 5.

When we evaluate a ² + b ² , we get 3 ² + 4 ² = 9 + 16 = 25.

The square of the hypotenuse (c ² ) is 5 ² = 25.

Since a ² + b ² equals c ² , it confirms that (3, 4, 5) is indeed a Pythagorean Triple.

These triples have applications in various fields, including geometry, trigonometry, and even number theory. They provide solutions to many practical problems involving right-angled triangles and are used in mathematics and engineering to calculate distances, angles, and more.

Pythagoras Triples Formula

The Pythagorean theorem precisely. When you have a right-angled triangle, where one angle is 90 degrees, you can establish a relationship between the lengths of its sides. In this context:

The longest side, opposite the right angle, is called the hypotenuse, and it's represented as 'r'.

The other two sides, which are adjacent to the right angle, are typically denoted as 'p' and 'q'.

The Pythagorean theorem states that the sum of the squares of the lengths of the two shorter sides ('p' and 'q') is equal to the square of the length of the hypotenuse ('r'), which can be expressed as:

p ² + q ² = r ²

Pythagorean Triples List

The list of Pythagorean triples where the value of c is above 100 is given below:

How to Form Pythagorean Triples?

Two common methods for generating Pythagorean triples based on whether the starting number is odd or even, and these methods work for many cases. However, it's important to note that these methods do not generate all Pythagorean triples.

For Odd Numbers (Case 1):

If 'x' is an odd number, you can create a Pythagorean triple using the formula: x, (x ² /2) – 0.5, (x ² /2) + 0.5.

For example, with x = 7, you get (7, 24, 25).

For Even Numbers (Case 2):

If 'x' is an even number, you can use the formula: x, (x/2) ² -1, (x/2) ² +1.

For instance, with x = 16, you get (16, 63, 65).

These methods can generate many Pythagorean triples, but they do not cover all possible triples. The reason is that Pythagorean triples can also be generated using coprime integers and other mathematical techniques. For example, the triple (20, 21, 29) doesn't fit into these methods, as you correctly pointed out.

To generate all possible Pythagorean triples, you would need more advanced mathematical techniques, such as Euclid's formula, which provides a systematic way to find all primitive (non-divisible by any common factor) Pythagorean triples. It's important to recognize that while these methods you've mentioned are useful, they only cover a subset of Pythagorean triples.

Triangular Numbers

A fascinating connection between squares, odd numbers, and triangular numbers. This connection demonstrates some interesting mathematical relationships:

Difference Between Successive Squares and Successive Odd Numbers:

The difference between successive squares (perfect squares) is indeed successive odd numbers. For example, the difference between 4 ² and 3 ² is 7 (which is an odd number), and the difference between 5 ² and 4 ² is 9 (also an odd number).

Squares as the Sum of Two Successive Triangular Numbers:

Every square can be expressed as the sum of two successive triangular numbers. Triangular numbers are the sums of consecutive positive integers, as you've mentioned.

For example, consider the square of 15 (15 ² = 225). You can express it as the sum of two successive triangular numbers: 105 (the 14th triangular number) and 120 (the 15th triangular number). So, 105 + 120 = 225, and 225 is the square of 15.

This relationship is an interesting illustration of how various mathematical concepts and sequences are interconnected. It also showcases the beauty and elegance of mathematics, where patterns and relationships can be found in seemingly unrelated areas of study.

Common Pythagorean Triples

Multiples of Pythagorean Triples:

You are correct that if (a, b, c) is a Pythagorean triple (i.e., it satisfies the Pythagorean theorem), then (ka, kb, kc) will also satisfy the theorem for any positive integer 'k'. In your example, you showed that by multiplying each integer in the triple (3, 4, 5) by 2, you get another valid Pythagorean triple (6, 8, 10). This property of Pythagorean triples is significant and allows for an infinite number of such triples.

Methods for Finding Pythagorean Triples:

You mentioned that there are various methods for finding Pythagorean triples, including using generalized Fibonacci sequences, quadratic equations, matrices, and linear transformations. These methods offer different ways to generate Pythagorean triples systematically.

Endless Set of Pythagorean Triples:

It's important to note that there is indeed an infinite number of Pythagorean triples, and this can be proven using the basic triple (3, 4, 5) as you mentioned. By generating new triples from existing ones (e.g., by multiplying by integers), you can create an unending sequence of Pythagorean triples.

Properties of Pythagorean Triples:

You correctly noted that Pythagorean triples typically consist of either all-even numbers or two odd numbers and an even number. This is due to the fact that the square of an odd number is always odd, and the square of an even number is always even. Consequently, for the Pythagorean theorem to hold, you'll have combinations that result in an even sum on one side (c ² ) and an odd sum on the other side (a ² + b ² ).

Pythagorean Triples Examples

(3, 4, 5):

This is one of the most famous Pythagorean triples.

It satisfies the Pythagorean theorem: 3 ² + 4 ² = 5 ² , where 9 + 16 = 25.

(5, 12, 13):

Another well-known Pythagorean triple.

It satisfies the Pythagorean theorem: 5 ² + 12 ² = 13 ² , where 25 + 144 = 169.

(7, 24, 25):

This triple is larger but still satisfies the Pythagorean theorem.

It satisfies the Pythagorean theorem: 7 ² + 24 ² = 25 ² , where 49 + 576 = 625.

(8, 15, 17):

A Pythagorean triple with relatively small numbers.

It satisfies the Pythagorean theorem: 8 ² + 15 ² = 17 ² , where 64 + 225 = 289.

(20, 21, 29):

This triple has a relatively small difference between the two shorter sides.

It satisfies the Pythagorean theorem: 20 ² + 21 ² = 29 ² , where 400 + 441 = 841.

(9, 40, 41):

A Pythagorean triple where the difference between the two shorter sides is significant.

It satisfies the Pythagorean theorem: 9 ² + 40 ² = 41 ² , where 81 + 1600 = 1681.

These are just a few examples of Pythagorean triples.