Pythagorean Triplets

Pythagorean triplets

Consider the following

32 +42 = + =

= 52

The collection of numbers 3, 4 and 5 is known as Pythagorean triplet.

We also have

62 +82 = + =

= 102

making 6, 8 and 10 another example of such triplet.

Similarly,

52 + 122 = + =

=132.

The numbers 5, 12, 13 form another such triplet.

For any natural number m > 1, we have:

2m2+m2−12=m2+12.

So, 2m,m2 – 1 and m2 + 1 forms a Pythagorean triplet.

Example 2: Now, write a Pythagorean triplet whose smallest member is 8 using the above general form

Instructions

The smallest triplet is 8

Say, m2 – 1 = 8
Upon solving we get, m =
Thus, m2+ 1 = and 2m =
This gives us the Pythagorean triplet of 6,8,10. However, as it turns out 8 isn't the smallest member. So, let's try another assumption.
Let's take 2m = 8
Which gives m =
Thus, m2+ 1 = and m2- 1 =
We have found the pythagoras triplet of 8,15 and 17 with 8 as the smallest member.

Example 3: Find a Pythagorean triplet in which one member is 12

Instructions

One triplet member is 12

Say, m2 – 1 = 12
Upon solving we find that: m isn't an integer. Let's try another assumption.
If we try m2 + 1 = 12, we get m2 = 11 which further gives a non-integer value for m
Thus, we can only take 2m = which gives m =.
Thus, m2+ 1 = and m2- 1 =
We have found the pythagoras triplet of 12,35 and 37.

This shows that our assumption when trying to find a suitable value of 'm' is important.

Note: All Pythagorean triplets may not be obtained using this form. For example: another triplet 5, 12, 13 also has 12 as a member.