FIO-10

Chapter: SQUARES AND SQUARE ROOTS
Class: 8 (Class 8)
Category: figure_it_out

Question

Are there primitive triples that cannot be obtained through this method? If yes, give examples.

The method discussed generates Pythagorean triples where the smallest number is always odd.

Step 1 — Understanding the method

Let us consider a common method to find Pythagorean triples. We start with an odd number, let's call it $x$. The three numbers in the triple are $x$, $\frac{x^2-1}{2}$, and $\frac{x^2+1}{2}$. Let us check if these numbers form a Pythagorean triple. This means we need to see if the square of the first number plus the square of the second number equals the square of the third number.

$x^2 + \left(\frac{x^2-1}{2}\right)^2$ $= x^2 + \frac{(x^2-1)^2}{4}$ $= \frac{4x^2 + (x^4 - 2x^2 + 1)}{4}$ $= \frac{x^4 + 2x^2 + 1}{4}$ $= \frac{(x^2+1)^2}{4}$ $= \left(\frac{x^2+1}{2}\right)^2$ This shows that the numbers $x$, $\frac{x^2-1}{2}$, and $\frac{x^2+1}{2}$ always form a Pythagorean triple.

Step 2 — Properties of triples from this method

Let us look at the numbers generated by this method. We start by choosing $x$ as an odd number. For example, if we choose $x = 3$: The numbers are $3$, $\frac{3^2-1}{2} = \frac{8}{2} = 4$, and $\frac{3^2+1}{2} = \frac{10}{2} = 5$. The triple is (3, 4, 5). The smallest number in this triple is 3, which is odd.

If we choose $x = 5$: The numbers are $5$, $\frac{5^2-1}{2} = \frac{24}{2} = 12$, and $\frac{5^2+1}{2} = \frac{26}{2} = 13$. The triple is (5, 12, 13). The smallest number in this triple is 5, which is odd.

For any odd number $x$ greater than 1, $x^2$ is also odd. So, $x^2-1$ will be an even number. This means $\frac{x^2-1}{2}$ will always be a whole number. Also, for $x \ge 3$, we can see that $x < \frac{x^2-1}{2}$. For example, if $x=3$, then $3 < 4$. So, the smallest number in the triple $(x, \frac{x^2-1}{2}, \frac{x^2+1}{2})$ is always $x$. Since we chose $x$ to be an odd number, the smallest number in any triple generated by this method is always odd.

Step 3 — Primitive Pythagorean triples

A primitive Pythagorean triple is a set of three whole numbers $(a, b, c)$ where $a^2 + b^2 = c^2$, and $a, b, c$ do not share any common factor other than 1. For example, (3, 4, 5) is primitive, but (6, 8, 10) is not primitive because all numbers are divisible by 2.

In any primitive Pythagorean triple, one of the smaller numbers (legs) must be odd, and the other must be even. It is not possible for both legs to be even (because then all three numbers would be even, making the triple not primitive). It is also not possible for both legs to be odd (because the sum of two odd squares would be a number that cannot be a perfect square).

The method from Step 1 always generates a primitive Pythagorean triple if $x$ is an odd number. This is because an odd number $x$ and $\frac{x^2-1}{2}$ will not share any common factors other than 1.

Step 4 — Triples not generated by this method

We are looking for primitive triples that cannot be obtained by the method discussed. Since the method discussed always produces a primitive triple where the smallest number is odd, we need to find primitive triples where the smallest number is even.

Let us look at some primitive Pythagorean triples:

• (3, 4, 5): Smallest number is 3 (odd). This triple can be generated by the method (with $x=3$).
• (5, 12, 13): Smallest number is 5 (odd). This triple can be generated by the method (with $x=5$).
• (8, 15, 17): Smallest number is 8 (even). Let us check if it is primitive: The numbers 8, 15, and 17 do not have any common factors other than 1. So, it is a primitive triple. Since its smallest number (8) is even, it cannot be generated by the method that always produces an odd smallest number.
• (12, 35, 37): Smallest number is 12 (even). Let us check if it is primitive: The numbers 12, 35, and 37 do not have any common factors other than 1. So, it is a primitive triple. Since its smallest number (12) is even, it cannot be generated by the method.
• (16, 63, 65): Smallest number is 16 (even). Let us check if it is primitive: The numbers 16, 63, and 65 do not have any common factors other than 1. So, it is a primitive triple. Since its smallest number (16) is even, it cannot be generated by the method.

Answer

Yes, there are primitive triples that cannot be obtained through this method.

(i) Examples of such triples are (8, 15, 17). (ii) Another example is (16, 63, 65).