Pythagorean Triplets

To generate 100 Pythagorean triplets, I'll use the Euclidean formula for primitive Pythagorean triplets, where for positive integers \( m \) and \( n \) with \( m > n \), the triplet is given by:

- \( a = m^2 - n^2 \)

- \( b = 2mn \)

- \( c = m^2 + n^2 \)

To ensure a diverse set of triplets, I'll focus on primitive triplets (where \( a \), \( b \), and \( c \) are coprime) by choosing \( m \) and \( n \) such that they are coprime and of opposite parity (one even, one odd). I'll also include some non-primitive triplets by scaling primitive ones to reach 100 unique triplets. Below is a systematic approach to generate them.

### Approach

1. **Generate Primitive Triplets**: Iterate over pairs of \( m \) and \( n \) (with \( m > n \), coprime, opposite parity) to produce primitive triplets.

2. **Scale for Non-Primitive Triplets**: Multiply primitive triplets by small scalars (e.g., 2, 3) to generate additional triplets.

3. **Ensure 100 Triplets**: Continue generating until at least 100 unique triplets are obtained, ensuring \( a \leq b < c \).

### List of 100 Pythagorean Triplets

Below are 100 Pythagorean triplets, starting with primitive ones and including scaled versions where needed. Each triplet \( (a, b, c) \) satisfies \( a^2 + b^2 = c^2 \). For readability, I'll list them in a compact format, grouped by the \( m \) and \( n \) values used to generate primitives, followed by scaled versions.

#### Primitive Triplets (Generated with \( m \), \( n \) coprime, opposite parity)

1. \( m=2, n=1 \): (3, 4, 5)

2. \( m=3, n=2 \): (5, 12, 13)

3. \( m=4, n=1 \): (15, 8, 17)

4. \( m=4, n=3 \): (7, 24, 25)

5. \( m=5, n=2 \): (21, 20, 29)

6. \( m=5, n=4 \): (9, 40, 41)

7. \( m=6, n=1 \): (35, 12, 37)

8. \( m=6, n=5 \): (11, 60, 61)

9. \( m=7, n=2 \): (45, 28, 53)

10. \( m=7, n=4 \): (33, 56, 65)

11. \( m=7, n=6 \): (13, 84, 85)

12. \( m=8, n=1 \): (63, 16, 65)

13. \( m=8, n=3 \): (55, 48, 73)

14. \( m=8, n=5 \): (39, 80, 89)

15. \( m=8, n=7 \): (15, 112, 113)

16. \( m=9, n=2 \): (77, 36, 85)

17. \( m=9, n=4 \): (65, 72, 97)

18. \( m=9, n=8 \): (17, 144, 145)

19. \( m=10, n=1 \): (99, 20, 101)

20. \( m=10, n=3 \): (91, 60, 109)

21. \( m=10, n=7 \): (51, 140, 149)

22. \( m=10, n=9 \): (19, 180, 181)

23. \( m=11, n=2 \): (117, 44, 125)

24. \( m=11, n=4 \): (105, 88, 137)

25. \( m=11, n=6 \): (85, 132, 157)

26. \( m=11, n=8 \): (57, 176, 185)

27. \( m=11, n=10 \): (21, 220, 221)

28. \( m=12, n=1 \): (143, 24, 145)

29. \( m=12, n=5 \): (119, 120, 169)

30. \( m=12, n=7 \): (95, 168, 193)

31. \( m=12, n=11 \): (23, 264, 265)

32. \( m=13, n=2 \): (165, 52, 173)

33. \( m=13, n=4 \): (153, 104, 185)

34. \( m=13, n=6 \): (133, 156, 205)

35. \( m=13, n=8 \): (105, 208, 233)

36. \( m=13, n=10 \): (69, 260, 269)

37. \( m=13, n=12 \): (25, 312, 313)

38. \( m=14, n=1 \): (195, 28, 197)

39. \( m=14, n=3 \): (187, 84, 205)

40. \( m=14, n=5 \): (171, 140, 221)

41. \( m=14, n=9 \): (115, 252, 277)

42. \( m=14, n=11 \): (75, 308, 317)

43. \( m=14, n=13 \): (27, 364, 365)

44. \( m=15, n=2 \): (221, 60, 229)

45. \( m=15, n=4 \): (209, 120, 241)

46. \( m=15, n=8 \): (161, 240, 289)

47. \( m=15, n=14 \): (29, 420, 421)

48. \( m=16, n=1 \): (255, 32, 257)

49. \( m=16, n=3 \): (247, 96, 265)

50. \( m=16, n=5 \): (231, 160, 281)

#### Scaled Triplets (Non-Primitive, by multiplying primitive triplets)

To reach 100 triplets, I'll scale some of the above primitive triplets by factors like 2, 3, or 4, ensuring no duplicates. For example, scaling (3, 4, 5) by 2 gives (6, 8, 10).

51. \( 2 \times (3, 4, 5) \): (6, 8, 10)

52. \( 3 \times (3, 4, 5) \): (9, 12, 15)

53. \( 4 \times (3, 4, 5) \): (12, 16, 20)

54. \( 2 \times (5, 12, 13) \): (10, 24, 26)

55. \( 3 \times (5, 12, 13) \): (15, 36, 39)

56. \( 2 \times (15, 8, 17) \): (30, 16, 34)

57. \( 3 \times (15, 8, 17) \): (45, 24, 51)

58. \( 2 \times (7, 24, 25) \): (14, 48, 50)

59. \( 3 \times (7, 24, 25) \): (21, 72, 75)

60. \( 2 \times (21, 20, 29) \): (42, 40, 58)

61. \( 2 \times (9, 40, 41) \): (18, 80, 82)

62. \( 2 \times (35, 12, 37) \): (70, 24, 74)

63. \( 2 \times (11, 60, 61) \): (22, 120, 122)

64. \( 2 \times (45, 28, 53) \): (90, 56, 106)

65. \( 2 \times (33, 56, 65) \): (66, 112, 130)

66. \( 2 \times (13, 84, 85) \): (26, 168, 170)

67. \( 2 \times (63, 16, 65) \): (126, 32, 130)

68. \( 2 \times (55, 48, 73) \): (110, 96, 146)

69. \( 2 \times (39, 80, 89) \): (78, 160, 178)

70. \( 2 \times (15, 112, 113) \): (30, 224, 226)

71. \( 2 \times (77, 36, 85) \): (154, 72, 170)

72. \( 2 \times (65, 72, 97) \): (130, 144, 194)

73. \( 2 \times (17, 144, 145) \): (34, 288, 290)

74. \( 2 \times (99, 20, 101) \): (198, 40, 202)

75. \( 2 \times (91, 60, 109) \): (182, 120, 218)

76. \( 2 \times (51, 140, 149) \): (102, 280, 298)

77. \( 2 \times (19, 180, 181) \): (38, 360, 362)

78. \( 2 \times (117, 44, 125) \): (234, 88, 250)

79. \( 2 \times (105, 88, 137) \): (210, 176, 274)

80. \( 2 \times (85, 132, 157) \): (170, 264, 314)

81. \( 2 \times (57, 176, 185) \): (114, 352, 370)

82. \( 2 \times (21, 220, 221) \): (42, 440, 442)

83. \( 2 \times (143, 24, 145) \): (286, 48, 290)

84. \( 2 \times (119, 120, 169) \): (238, 240, 338)

85. \( 2 \times (95, 168, 193) \): (190, 336, 386)

86. \( 2 \times (23, 264, 265) \): (46, 528, 530)

87. \( 2 \times (165, 52, 173) \): (330, 104, 346)

88. \( 2 \times (153, 104, 185) \): (306, 208, 370)

89. \( 2 \times (133, 156, 205) \): (266, 312, 410)

90. \( 2 \times (105, 208, 233) \): (210, 416, 466)

91. \( 2 \times (69, 260, 269) \): (138, 520, 538)

92. \( 2 \times (25, 312, 313) \): (50, 624, 626)

93. \( 2 \times (195, 28, 197) \): (390, 56, 394)

94. \( 2 \times (187, 84, 205) \): (374, 168, 410)

95. \( 2 \times (171, 140, 221) \): (342, 280, 442)

96. \( 2 \times (115, 252, 277) \): (230, 504, 554)

97. \( 2 \times (75, 308, 317) \): (150, 616, 634)

98. \( 2 \times (27, 364, 365) \): (54, 728, 730)

99. \( 2 \times (221, 60, 229) \): (442, 120, 458)

100. \( 2 \times (209, 120, 241) \): (418, 240, 482)

### Verification

Each triplet satisfies \( a^2 + b^2 = c^2 \). For example:

- For (3, 4, 5): \( 3^2 + 4^2 = 9 + 16 = 25 = 5^2 \)

- For (418, 240, 482): \( 418^2 + 240^2 = 174724 + 57600 = 232324 = 482^2 \)

The list includes 50 primitive triplets (from \( m=2 \) to \( m=16 \)) and 50 non-primitive triplets (scaled by 2 or higher). All are unique and valid.

### Notes

- The triplets are presented with \( a \leq b < c \) where possible, though the formula may generate \( b < a \); swapping \( a \) and \( b \) doesn't affect validity.

- If you need these in a specific format (e.g., sorted by \( c \), or as a CSV file), or a program to generate them, let me know!

- If you want a chart visualizing some property of these triplets (e.g., distribution of \( c \)), I can create one upon confirmation.

Would you like any further processing, such as a program to generate these, or a chart?