Exploring Pythagorean Triples with Members Less Than 100

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. These triples have captivated mathematicians for centuries due to their elegance and utility in various fields, including geometry, number theory, and even cryptography.

Primitive Pythagorean Triples

Before delving into the numbers, it's important to understand the concept of primitive Pythagorean triples. A primitive Pythagorean triple consists of three positive integers where the greatest common divisor (GCD) is 1, meaning the numbers have no common factors other than 1. In simpler terms, a primitive Pythagorean triple is a Pythagorean triple that cannot be obtained by multiplying another Pythagorean triple by any integer greater than 1.

Listing of Primitive Pythagorean Triples

Here is a list of primitive Pythagorean triples where all three members are less than 100. These sets of numbers can be used to form right-angled triangles with all side lengths being integers:

(3, 4, 5): (3^2 4^2 5^2) (5, 12, 13): (5^2 12^2 13^2) (7, 24, 25): (7^2 24^2 25^2) (8, 15, 17): (8^2 15^2 17^2) (9, 40, 41): (9^2 40^2 41^2) (11, 60, 61): (11^2 60^2 61^2) (12, 35, 37): (12^2 35^2 37^2) (13, 84, 85): (13^2 84^2 85^2) (16, 63, 65): (16^2 63^2 65^2) (20, 21, 29): (20^2 21^2 29^2) (28, 45, 53): (28^2 45^2 53^2) (33, 56, 65): (33^2 56^2 65^2) (36, 77, 85): (36^2 77^2 85^2) (39, 80, 89): (39^2 80^2 89^2) (48, 55, 73): (48^2 55^2 73^2) (65, 72, 97): (65^2 72^2 97^2)

Generating More Pythagorean Triples

While the list provided is extensive, it is not comprehensive as there are infinitely many Pythagorean triples. However, all primitive Pythagorean triples can be generated using the formula:

This formula generates Pythagorean triples:

Let ( m > n > 0 ) be an odd integer and ( N frac{m^2 n^2}{2} ) Then the Pythagorean triple is given by ( (N - n, N, N n) )

By iterating over different values of ( m ) and ( n ), we can generate an infinite number of Pythagorean triples. For example, with ( m 5 ) and ( n 1 ), we get the triple (12, 35, 37).

Conclusion

The largest known primitive Pythagorean triple with all three members less than 100 is (65, 72, 97). This means the hypotenuse 97 is the largest side that still allows the other two sides to be integers within 100. This concept is not only a fascinating mathematical curiosity but also serves as a practical tool in various applications, such as solving problems in geometry or cryptography.

For those interested in exploring more about Pythagorean triples, resources like Jason Chu's list, which contains 50 exact triples, provide additional insight and detail. Remember, while these triples are useful, they are fundamental in understanding the broader implications of the Pythagorean theorem.