Question 275017: Pythagorean triplets represent a special relation that both satisfy the Pythagorean theorem and as well as another characteristic. It is this additional characteristic that I am especially looking for. There are several formulas that will produce Pythagorean triplets. I need to find at least one. I need at least 5 other examples of triplets, not including 3-4-5.
Found 2 solutions by Edwin McCravy, Alan3354:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website! Pythagorean triplets represent a special relation that both satisfy the Pythagorean theorem and as well as another characteristic. It is this additional characteristic that I am especially looking for. There are several formulas that will produce Pythagorean triplets. I need to find at least one. I need at least 5 other examples of triplets, not including 3-4-5.
One formula is
where you can substitute any positive integers for  and  .
if k = 1 and q = 1 then we have a = 3, b = 4; and c = 5
if k = 1 and q = 2 then we have a = 5, b = 12; and c = 13
if k = 1 and q = 3 then we have a = 7, b = 24; and c = 25
if k = 1 and q = 4 then we have a = 9, b = 40; and c = 41
if k = 1 and q = 5 then we have a = 11, b = 60; and c = 61
if k = 1 and q = 6 then we have a = 13, b = 84; and c = 85
if k = 1 and q = 7 then we have a = 15, b = 112; and c = 113
if k = 1 and q = 8 then we have a = 17, b = 144; and c = 145
if k = 1 and q = 9 then we have a = 19, b = 180; and c = 181
if k = 1 and q = 10 then we have a = 21, b = 220; and c = 221
if k = 2 and q = 1 then we have a = 8, b = 6; and c = 10
if k = 2 and q = 2 then we have a = 12, b = 16; and c = 20
if k = 2 and q = 3 then we have a = 16, b = 30; and c = 34
if k = 2 and q = 4 then we have a = 20, b = 48; and c = 52
if k = 2 and q = 5 then we have a = 24, b = 70; and c = 74
if k = 2 and q = 6 then we have a = 28, b = 96; and c = 100
if k = 2 and q = 7 then we have a = 32, b = 126; and c = 130
if k = 2 and q = 8 then we have a = 36, b = 160; and c = 164
if k = 2 and q = 9 then we have a = 40, b = 198; and c = 202
if k = 2 and q = 10 then we have a = 44, b = 240; and c = 244
if k = 3 and q = 1 then we have a = 15, b = 8; and c = 17
if k = 3 and q = 2 then we have a = 21, b = 20; and c = 29
if k = 3 and q = 3 then we have a = 27, b = 36; and c = 45
if k = 3 and q = 4 then we have a = 33, b = 56; and c = 65
if k = 3 and q = 5 then we have a = 39, b = 80; and c = 89
if k = 3 and q = 6 then we have a = 45, b = 108; and c = 117
if k = 3 and q = 7 then we have a = 51, b = 140; and c = 149
if k = 3 and q = 8 then we have a = 57, b = 176; and c = 185
if k = 3 and q = 9 then we have a = 63, b = 216; and c = 225
if k = 3 and q = 10 then we have a = 69, b = 260; and c = 269
if k = 4 and q = 1 then we have a = 24, b = 10; and c = 26
if k = 4 and q = 2 then we have a = 32, b = 24; and c = 40
if k = 4 and q = 3 then we have a = 40, b = 42; and c = 58
if k = 4 and q = 4 then we have a = 48, b = 64; and c = 80
if k = 4 and q = 5 then we have a = 56, b = 90; and c = 106
if k = 4 and q = 6 then we have a = 64, b = 120; and c = 136
if k = 4 and q = 7 then we have a = 72, b = 154; and c = 170
if k = 4 and q = 8 then we have a = 80, b = 192; and c = 208
if k = 4 and q = 9 then we have a = 88, b = 234; and c = 250
if k = 4 and q = 10 then we have a = 96, b = 280; and c = 296
if k = 5 and q = 1 then we have a = 35, b = 12; and c = 37
if k = 5 and q = 2 then we have a = 45, b = 28; and c = 53
if k = 5 and q = 3 then we have a = 55, b = 48; and c = 73
if k = 5 and q = 4 then we have a = 65, b = 72; and c = 97
if k = 5 and q = 5 then we have a = 75, b = 100; and c = 125
if k = 5 and q = 6 then we have a = 85, b = 132; and c = 157
if k = 5 and q = 7 then we have a = 95, b = 168; and c = 193
if k = 5 and q = 8 then we have a = 105, b = 208; and c = 233
if k = 5 and q = 9 then we have a = 115, b = 252; and c = 277
if k = 5 and q = 10 then we have a = 125, b = 300; and c = 325
if k = 6 and q = 1 then we have a = 48, b = 14; and c = 50
if k = 6 and q = 2 then we have a = 60, b = 32; and c = 68
if k = 6 and q = 3 then we have a = 72, b = 54; and c = 90
if k = 6 and q = 4 then we have a = 84, b = 80; and c = 116
if k = 6 and q = 5 then we have a = 96, b = 110; and c = 146
if k = 6 and q = 6 then we have a = 108, b = 144; and c = 180
if k = 6 and q = 7 then we have a = 120, b = 182; and c = 218
if k = 6 and q = 8 then we have a = 132, b = 224; and c = 260
if k = 6 and q = 9 then we have a = 144, b = 270; and c = 306
if k = 6 and q = 10 then we have a = 156, b = 320; and c = 356
if k = 7 and q = 1 then we have a = 63, b = 16; and c = 65
if k = 7 and q = 2 then we have a = 77, b = 36; and c = 85
if k = 7 and q = 3 then we have a = 91, b = 60; and c = 109
if k = 7 and q = 4 then we have a = 105, b = 88; and c = 137
if k = 7 and q = 5 then we have a = 119, b = 120; and c = 169
if k = 7 and q = 6 then we have a = 133, b = 156; and c = 205
if k = 7 and q = 7 then we have a = 147, b = 196; and c = 245
if k = 7 and q = 8 then we have a = 161, b = 240; and c = 289
if k = 7 and q = 9 then we have a = 175, b = 288; and c = 337
if k = 7 and q = 10 then we have a = 189, b = 340; and c = 389
if k = 8 and q = 1 then we have a = 80, b = 18; and c = 82
if k = 8 and q = 2 then we have a = 96, b = 40; and c = 104
if k = 8 and q = 3 then we have a = 112, b = 66; and c = 130
if k = 8 and q = 4 then we have a = 128, b = 96; and c = 160
if k = 8 and q = 5 then we have a = 144, b = 130; and c = 194
if k = 8 and q = 6 then we have a = 160, b = 168; and c = 232
if k = 8 and q = 7 then we have a = 176, b = 210; and c = 274
if k = 8 and q = 8 then we have a = 192, b = 256; and c = 320
if k = 8 and q = 9 then we have a = 208, b = 306; and c = 370
if k = 8 and q = 10 then we have a = 224, b = 360; and c = 424
if k = 9 and q = 1 then we have a = 99, b = 20; and c = 101
if k = 9 and q = 2 then we have a = 117, b = 44; and c = 125
if k = 9 and q = 3 then we have a = 135, b = 72; and c = 153
if k = 9 and q = 4 then we have a = 153, b = 104; and c = 185
if k = 9 and q = 5 then we have a = 171, b = 140; and c = 221
if k = 9 and q = 6 then we have a = 189, b = 180; and c = 261
if k = 9 and q = 7 then we have a = 207, b = 224; and c = 305
if k = 9 and q = 8 then we have a = 225, b = 272; and c = 353
if k = 9 and q = 9 then we have a = 243, b = 324; and c = 405
if k = 9 and q = 10 then we have a = 261, b = 380; and c = 461
if k = 10 and q = 1 then we have a = 120, b = 22; and c = 122
if k = 10 and q = 2 then we have a = 140, b = 48; and c = 148
if k = 10 and q = 3 then we have a = 160, b = 78; and c = 178
if k = 10 and q = 4 then we have a = 180, b = 112; and c = 212
if k = 10 and q = 5 then we have a = 200, b = 150; and c = 250
if k = 10 and q = 6 then we have a = 220, b = 192; and c = 292
if k = 10 and q = 7 then we have a = 240, b = 238; and c = 338
if k = 10 and q = 8 then we have a = 260, b = 288; and c = 388
if k = 10 and q = 9 then we have a = 280, b = 342; and c = 442
if k = 10 and q = 10 then we have a = 300, b = 400; and c = 500
Edwin
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! In addition to the extensive list provided by the other tutor, it's interesting that
But, there are no instances of integers that fit
 for any n <> 2
This was recently proven after centuries of effort - Fermat's Last Theorem.
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