An Introduction
Let’s get started with the basics by knowing what Pythagorean triples are.
According to Pythagoras theorem, the square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the other two sides.
The Pythagoras theorem can be usually expressed as p2+q2 = r2
Now, when we replace the variables p, q, r with some integers and if they satisfy the above equation then they are known as Pythagorean triples.
The most common examples of Pythagorean triples are (3, 4 , 5) and (5,12,13).
Now, you might think that the Pythagorean Theorem can be used for any triangle.
But the theorem is applicable only for a right-angle triangle.
What is a Right-Angle Triangle?
To understand what a right angle triangle is, let us consider a right-angle triangle named ABC, with its three sides namely the opposite, adjacent, and the hypotenuse. In a right-angled triangle, we generally refer to the three sides according to their relation with the angle. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.
The three sides of a right-angled triangle are as follows-
The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse (H).
The side that is opposite to the angle θ is known as the opposite (O).
The side which lies next to the angle is known as the Adjacent(A)
Pythagoras theorem states that,
In any right-angled triangle,
(Opposite)2 + (Adjacent)2 = (Hypotenuse)2
Something you need to know!
When we multiply the entries in a triple by any integer and get another Pythagorean triple.
For example :- (6 , 8,10), (9,12,15) and (15,20,25).
How to find Pythagorean Triples?
Here are the rules of how to find Pythagorean Triples,
Each and every odd number is the p side of a Pythagorean triplet( p2+q2 = r2)
The q side in a Pythagorean triplet is equally to (p2 – 1)/2.
The r side is equal to q + 1.
Now, we know that p and r are always odd and q is even.
These relationships are true because the difference between successive square numbers is the successive odd numbers.
All the odd numbers in the relationship are themselves a square (and the square of all odd numbers is an odd number itself) which gives us a Pythagorean triplet.
Primitive Pythagorean Triples:
The triples for which the entries are relatively prime are known as Primitive Pythagorean Triples. But all three legs in the Primitive Pythagorean triples cannot be prime.
As we can see in the common example (3, 4, 5), the numbers 3 and 5 are prime numbers and here 4 is an even number.
So in any given primitive Pythagorean triple one of the three entries must be even.
The most elementary number theory texts prove that all primitive triples (p,q, r) are given by the following:
p = u2 – v2, q= 2uv, r = u2 + v2
In the above equation u and v are relatively prime integers, not both odd. We observe that p is a difference of squares, so for it to be prime we need that u and v differ by 1. So, we can write,
a = 2v + 1, b = 2v2 + 2v, and c = 2v2 + 2v + 1.
Here are the first few triples we then expect to see infinitely many triples with two primes according to Schinzel and Sierpinski's Hypothesis H.
The Pythagorean triples list consists of all the possible Pythagorean triples.
Pythagorean Triplets list:
These are some Pythagorean triples examples.
Questions to be Solved :
1. Which of the following is not a Pythagorean triple?
a) 7, 24, 25
b) 8, 15, 17
c) 9, 12, 15
d)10,16,19
Ans: Pythagorean Triples are sets of whole numbers which fit the rule:
a2 + b2 = c2
In Option a): 72 + 242 = 49 + 576 = 625 = 252 which means 7, 24 and 25 is a Pythagorean triple.
In Option b): 82 + 152 = 64 + 225 = 289 = 172 which means 8, 15, and 17 is a Pythagorean triple.
In Option c): 92 + 122 = 81 + 144 = 225 = 152 which means 9, 12 and 15 is a Pythagorean triple.
In Option d): 102 + 162 = 100 + 256 = 356 ≠ 192 which means 10, 16 and 19 is not a Pythagorean triple.
(7, 24, 25 ),( 8, 15, 17), (9, 12, 15) are Pythagorean triple examples.
2. Check if (8, 15,17) are Pythagorean triples?
Ans: Given, Pythagorean triples = (8, 15, 17)
We can say that p = 17, q = 15, r = 8
We know the Pythagorean triples formula is, p2 = q2 + r2
LHS, p2 = 172 = 289
RHS, r2 + q2 = 82 + 152
= 64 +225 = 289
We see that in the given question,
LHS = RHS
Therefore, the given number set (7, 24, 25) is one of the Pythagorean triple examples.
FAQs on Pythagorean Triples
1. What are examples of Pythagorean Triples and what are the smallest Pythagorean Triples?
The Pythagorean theorem has been derived from the Pythagorean triples proof which states that integer triples which satisfy this equation are known as Pythagorean triples. The most common examples are (3,4,5) and (5,12,13) that are very common in Mathematics. Notice that when we multiply the entries in a triple by any integer, we get another Pythagorean triple. For example, (6, 8,10), (9,12,15) and (15,20,25).The smallest Pythagorean Triple in Mathematics is 3, 4 and 5 in Mathematics.
2. What is a Pythagorean Triple?
A Pythagorean triple is made up of three positive numbers a, b, and c that add up to c2 when a2 + b2 = c2. A typical way to write such a triple is (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is considered to be a Pythagorean triple, then (ka, kb, kc) is also a Pythagorean triple for a positive integer k. A primitive Pythagorean triple has three coprime numbers: a, b, and c. A Pythagorean triangle is described as a triangle whose sides form a Pythagorean triple and is always known to be a right triangle.
Pythagorean triples are named after the Pythagorean theorem, which states that any right triangle has side lengths that satisfy the formula a2 + b2 = c2; consequently, Pythagorean triples describe a right triangle's three integer side lengths. Right triangles with non-integer sides, on the other hand, do not generate Pythagorean triples.
Since ancient times, Pythagorean triples have been known. Plimpton 322, a Babylonian clay tablet written in a sexagesimal numeral system from around 1800 BC, is the oldest known record. Edgar James Banks discovered it shortly after 1900 and sold it to George Arthur Plimpton for $10 in 1922. [2]
The equation a2 + b2 = c2 is a Diophantine equation when looking for integer solutions. Pythagorean triples are thus among the oldest known nonlinear Diophantine problem solutions. Click here to learn more.
3. How to generate a Triple Pythagorean?
Given any pair of integers m and n with m > n > 0, Euclid's formula is a fundamental formula for creating Pythagorean triples. According to the formula, the integers:
a = m2 - n2, b = 2mn, c = m2 + n2
A Pythagorean triple is formed. If and only if m and n are coprime and not both odd, the triple formed by Euclid's formula is primitive. In a scenario where both n and m are odd, a, b, and c will be even, and the triple will not be primitive. But when m and n are coprime and both odd, dividing a, b, and c by 2 will provide a primitive triple.
Every primitive triple is formed by a unique pair of coprime numbers m, n, one of which is even (after the exchange of a and b, if an is even). As a result, there are an endless number of primordial Pythagorean triples. Throughout the rest of this article, the relationship of a, b, and c to m and n from Euclid's formula is referenced.
In Spite of the fact that Euclid's formula generates all primitive triples, it does not yield all triples. For example - 9, 12, 15 cannot be produced by making use of integers m and n. This can be fixed by adding an extra k parameter to the formula. All Pythagorean triples will be generated in this way:
a = k. (m2 - n2), b = k. (2mn), c = k. (m2 + n2)
where m, n, and k are positive integers with m > n, m and n coprime, and m and n not both odd.
The argument that these formulas generate Pythagorean triples can be verified by extending a2 + b2 with foundational algebra and verifying that the result equals c2. As every Pythagorean triple can be divided by some integer k to get a primitive triple, each triple can be made once only by using the formula with m and n to construct its primitive counterpart and then multiplying by k as in the last equation.
Choosing m and n from a variety of integer sequences yields some intriguing results. If m and n are successive Pell numbers, for example, a and b will differ by one. Choosing m and n from a variety of integer sequences yields some intriguing results. If m and n are successive Pell numbers, for example, a and b will differ by one.
Since Euclid's time, many formulas for creating triples with specific qualities have been discovered.
4. Write a note on the distribution of Pythagorean Triples.
The distribution of Pythagorean triples has a variety of results. A number of evident patterns can already be seen in the scatter plot. The legs of a primitive triple appear in the plot. Any integer multiples of must also co-exist in the plot.
There are sets of parabolic patterns with a high density of points and all of their foci at the origin that open up in all four directions within the dispersion. Various parabolas intersect at the axes and appear to reflect off the axis with a 45-degree incidence angle, with a third parabola making an entry perpendicularly.
5. What are the heronian triangle triples?
We will look at Heronian triangles with distinct integer sides. A Heronian triangle is typically defined as one with integer sides and an integer area. The lengths of the triangle's sides create a Heronian triple (a, b, c) if a, b, and c are specified. Every Pythagorean triple can be called a Heronian triple. As in a Pythagorean triple, at least one of the legs a, b must be even, hence the area ab/2 is an integer. As the example (4, 13, 15) with area 24 reveals, not every Heronian triple is a Pythagorean triple.
6. What roles do the parabolas play in Pythagorean Triples?
The angular properties of the parabolas are instantly apparent from their functional form. At a = 2n, the parabolas are mirrored at the a-axis, and the derivative of b with respect to an is –1, resulting in a 45° incidence angle. The value 2n also corresponds to a cluster since the clusters, like other triples, are repeated at integer multiples. When a and b are interchanged, the corresponding parabola crosses the b-axis at right angles at b = 2n, and thus its reflection meets the a-axis at right angles at a = 2n, exactly in the position where the parabola for n is reflected at the a-axis. The Vedantu app and website contain free study materials.
