In this blog, we will discuss Pythagorean triple or Pythagorean triplets .
What is a Pythagorean Triple
A Pythagorean triple has three positive integers a, b and c, such that a2+b2=c2
Consider the following
32+ 42= 9 + 16 = 25 = 52 The collection of numbers 3, 4 and 5 is known as Pythagorean triplet.
6, 8, 10 is also a Pythagorean triplet, since
62 + 82= 36 + 64 = 100 = 102
Again, observe that
52+ 122= 25 + 144 = 169 = 132. The numbers 5, 12, 13 form another such triplet.
What is Pythagorean triple?
For any natural number m > 1, we have (2m)2 + (m2 - 1) 2 =/ (m2 + 1) 2. So, 2m, m2 - 1 and m2+ 1 form a Pythagorean triplet.
Example: Write a Pythagorean triplet whose smallest member is 8.
Solution: Let us first take m2 - 1 = 8
So, m2 = 8 + 1 = 9 which gives m = 3
Therefore, 2m = 6 and m2+ 1 = 10
The triplet is thus 6, 8, and 10. But 8 is not the smallest member of this.
So, let us try 2m = 8
m = 4
m2 - 1 = 16 - 1 = 15 and m2+ 1 = 16 + 1 = 17
The triplet is 8, 15, and 17 with 8 as the smallest member.
Pythagorean Triple List
| (3, 4, 5) | (5, 12, 13) | (8, 15, 17) | (7, 24, 25) |
| (20, 21, 29) | (12, 35, 37) | (9, 40, 41) | (28, 45, 53) |
| (11, 60, 61) | (16, 63, 65) | (33, 56, 65) | (48, 55, 73) |
| (13, 84, 85) | (36, 77, 85) | (39, 80, 89) | (65, 72, 97) |
| (20, 99, 101) | (60, 91, 109) | (15, 112, 113) | (44, 117, 125) |
| (88, 105, 137) | (17, 144, 145) | (24, 143, 145) | (51, 140, 149) |
| (85, 132, 157) | (119, 120, 169) | (52, 165, 173) | (19, 180, 181) |
| (57, 176, 185) | (104, 153, 185) | (95, 168, 193) | (28, 195, 197) |
| (84, 187, 205) | (133, 156, 205) | (21, 220, 221) | (140, 171, 221) |
| (60, 221, 229) | (105, 208, 233) | (120, 209, 241) | (32, 255, 257) |
| (23, 264, 265) | (96, 247, 265) | (69, 260, 269) | (115, 252, 277) |
Read More: Properties of Square Numbers With Examples: Square and Square Roots
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