Some numbers, such as 3, 4, and 5, seem to function perfectly in the Pythagorean Theorem, as 32 + 42 = 52. Pythagorean triples are sets of positive, whole numbers that function in the Pythagorean Theorem. For three positive integers on a number line to be Pythagorean triples, they must work in the Pythagorean Theorem’s formula:
a2 + b2 = c2
A Pythagorean triple is a sequence of three positive integers that satisfy the Pythagorean theorem. The Pythagorean theorem demonstrates the relationship between the squares of the sides of any right triangle with a 90-degree (square) corner. The Pythagorean triples formula, which is a set of positive integers that satisfy the Pythagorean theorem, will be discussed here or you can check the Cuemath website. Cuemath is an online learning platform that helps students explore the why behind what of every concept. Cuemath uses interesting learning tools that keep students motivated and engaged.
Pythagorean triples are relatively prime. Relatively prime states that they have no common divisor other than 1, even if the numbers are not prime numbers, like 14 and 15. The only common factor is 1.
Primitive Pythagorean Triples
The Pythagorean triples formula is made up of three integers that obey the Pythagorean theorem’s laws. Pythagorean triples are a type of triple that is commonly written in the form of (a,b,c). The Pythagorean triangle is the triangle created by these triples.
A set of numbers are considered to be a primitive Pythagorean triple if the three numbers have no common divisor other than number 1 from the entire number system. Since (3, 4, and 5) have no common divisors other than 1, our first Pythagorean triple is primitive. However, since each value for a, b, and c of the right triangle is a multiple of 5, our fifth set from the previous example is not primitive.
Non – Primitive Pythagorean Triples
A set of numbers that are considered to be a non-primitive Pythagorean triple is if all the three numbers in the triples have a common divisor in the sequence.
How To Find the Pythagorean Triples
Here are a few tips on how to find Pythagorean triples in three simple steps:
- Pick an even number to behave as the longer leg’s length.
- Then find a prime number that is one greater than that even number, to behave as the hypotenuse.
- Finally, calculate the third value to find the Pythagorean triple.
Pythagorean Triples Formula with an example:
Suppose one picks 12 as the length of a leg, knowing 13 is an adjacent prime number of number 12. Now use these two numbers as a part of the Pythagorean Theorem to finish your primitive Pythagorean triple:
a2 + b2 = c2
a2 + 122 = 132
a2 + 144 = 169
Now subtract the value of b2 from both sides of the equation:
a2 + (144-144) = (169-144)
So after solving this above equation our primitive Pythagorean triple is 5, 12, 13. One can simply generate iterations of that, none of which will be primitive by using the same multiplying method we used before: (10, 24, 26); (15, 36, 39); and many more.
Properties of Pythagorean Triples:
- Both even numbers, or two odd numbers and an even number, make up a Pythagorean Triple. The Pythagorean triples are formed when any constant number multiplies all of the numbers in a triplet.
- A Pythagorean Triple can never be made up of all odd numbers or two even numbers and one odd number since the square of an odd number is an odd number and the square of an even number is an even number, according to the properties of a square number.
- The sum of two even numbers is always an even number, and the sum of two odd numbers is always an odd number. When the values of a and b are both even, the value of c is also even. If one of the values of an or b is odd and the other is even, the value of c is also odd.