Pythagorean Triples

The integer solutions to the Pythagorean Theorem, a2 + b2 = c2 are called Pythagorean Triples.

You can say “triplets,” but “triples” are the favored term. Let’s start this topic by an introduction of Pythagoras theorem.

Pythagoras Theorem

Pythagoras who was a mathematician was interested in mathematics, science, and philosophy. He was born in Greece in about 570 BC. He is famous for a property of triangles with a right angle i.e 900 angles, and the property is known as Pythagoras Theorem.

If a triangle has one angle which is a right-angle (i.e. 90o), there exists a relationship between the three sides of the triangle.

If the longest side (called the hypotenuse) is r and the other two sides (next to the right angle) is called p and q, then:

p2 + q2 = r2.

Pythagoras’ Theorem

or,

The sum of the squares of the other two sides is the same as the square of the longest side.

In a right triangle, the hypotenuse is the side r, the side opposite the right angle. Adjacent to the right angle the shorter of the two sides is the side p.

The rules for determining a subset of Pythagorean triples

These rules are as below:

  • Every odd number is the p side of a Pythagorean triplet.
  • The q side of a Pythagorean triplet is simply \(\frac{(p2 – 1)}{2}\)
  • The r side is q + 1.

Now, p and r are always odd; q is even.

  • These relationships are true as the difference between successive square numbers is successive odd numbers.
  • All odd numbers which are itself a square (and the square of all odd number is an odd number itself) thus giving a Pythagorean triplet.

Pythagorean Triples Example:

  • So, the square of 3, 9, is the difference between 16, the square of 4, and 25 the square of 5, giving us the triplet 7,24,25.
  • Similarly, the square of 5, 25 is the difference between 144, the square of 12, and 169, the square of 13, giving us the triplet 5, 12, 13.

Pythagorean triples Proof:

Proof of Pythagoras theorem:

Look at the figure above

In the figure, at left,

Area of square = (p+q)2

Area of Triangle = \(\frac{1}{2}\) pq

Area of the inner square = q2.

The area of the entire square = 4(\(\frac{1}{2}\)pq) + r2

Now we can conclude that

(p + q)2 = 4(\(\frac{1}{2}\) pq) + r2.

or

p2 + 2pq + r2 = 2pq + r2.

Simplifying, we get Pythagorean triples formula,

p2 + q2 = r22

Hence Proved.

Triangular Numbers

The difference between successive squares is successive odd numbers is a fact and suggests that every square is the sum of two successive triangular numbers.

And in this, the triangular numbers are the successive sums of all integers.

  • 0 + 1 = 1,
  • 0 + 1 + 2 = 3,
  • 0 + 1 + 2 + 3 = 6, etc.

So the triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, etc.

105 + 120 = 225; 225 is the square of 15.

Pythagorean Triples list

(3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17)
(9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97)

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Practise This Question

Let x=pq be a rational number, such that the prime factorization of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is ____________.