Lecture Notes: Pythagorean Triples

Introduction to Pythagorean Triples

• Definition: Pythagorean triples are sets of three whole numbers that satisfy the equation of the Pythagorean theorem: (a^2 + b^2 = c^2).
• Example: The set (3, 4, 5) is a Pythagorean triple because (3^2 + 4^2 = 5^2).

Common Pythagorean Triples

1. (3, 4, 5) Triangle
   • Multiples: (6, 8, 10); (9, 12, 15); (12, 16, 20); (15, 20, 25).
2. (5, 12, 13) Triangle
   • Multiples: (10, 24, 26); (15, 36, 39).
3. (7, 24, 25) Triangle
4. (8, 15, 17) Triangle

• Note: These four sets are the most common and should be memorized for typical math courses.

Less Common Pythagorean Triples

• (9, 40, 41) Triangle

• (11, 60, 61) Triangle

• (12, 35, 37) Triangle

• (20, 21, 29) Triangle

• Note: These are less common but might appear in some problems.

Rare Pythagorean Triples

• Examples:

  • (13, 84, 85)
  • (16, 63, 65)
  • (28, 45, 53)
  • (33, 56, 65)
  • (36, 77, 85)
  • (39, 80, 89)
  • (48, 55, 73)
  • (65, 72, 97)

• Note: These are not typically needed for most problems but can be found online.

Pythagorean Triples Beyond 100

• Examples:
  • (15, 112, 113)
  • (17, 144, 145)
  • (19, 180, 181)
  • (20, 99, 101)
  • (21, 220, 221)
  • (23, 264, 265)
  • (24, 143, 145)
  • (25, 312, 313)

Conclusion

• Key Takeaway: Memorize the common and somewhat common Pythagorean triples (first eight in blue and red) for most problems. Additional rare triples are often unnecessary unless specified by a specific problem.