UPDATE 4/25/19: Greeting to all who have recently found this old blog post of mine. Welcome. It has not been updated since June 2012 (except for this notice), and it likely will not be any time soon. However, feel free to leave comments, especially if there's another primitive triple missing from the list.
| Leg | Primitive | Non-Primitive |
| 3 | 3,4,5 | -- |
| 4 | -- | -- |
| 5 | 5,12,13 | -- |
| 6 | -- | 6,8,10 |
| 7 | 7,24,25 | -- |
| 8 | 8,15,17 | -- |
| 9 | 9,40,41 | 9,12,15 |
| 10 | -- | 10,24,26 |
| 11 | 11,60,61 | -- |
| 12 | 12,35,37 | 12,16,20 |
| 13 | 13,84,85 | -- |
| 14 | -- | 14,48,50 |
| 15 | 15,112,113 | 15,20,25; 15,36,39 |
| 16 | 16,63,65 | 16,30,34 |
| 17 | 17,144,145 | -- |
| 18 | -- | 18,24,30; 18,80,82 |
| 19 | 19,180,181 | -- |
| 20 | 20,99,101; 20,21,29 | 20,48,52 |
| 21 | 21,220,221 | 21,28,35; 21,72,75 |
| 22 | -- | 22,120,122 |
| 23 | 23,264,265 | -- |
| 24 | 24,143,145 | 24,32,40; 24,45,51; 24,70,74 |
| 25 | 25,312,313 | 25,60,65 |
| 26 | -- | 26,168,170 |
| 27 | 27,364,365 | 27,36,45; 27,120,123 |
| 28 | 28,195,197 | 28,96,100 |
| 29 | 29,420,421 | -- |
| 30 | -- | 30,40,50; 30,72,78; 30,224,226 |
| 31 | 31,480,481 | -- |
| 32 | 32,255,257 | 32,60,68; 32,126,130 |
| 33 | 33,544,545 | 33,44,55; 33,180,183 |
| 34 | -- | 34,288,290 |
| 35 | 35,612,613 | 35,84,91; 35,120,125 |
| 36 | 36,323,325 | 36,48,60; 36,160,164; 36,105,111 |
| 37 | 37,684,685 | -- |
| 38 | -- | 38,360,362 |
| 39 | 39,760,761 | 39,52,65; 39,252,255 |
| 40 | 40,399,401 | 40,96,104; 40,75,85; 40,198,202; 40,42,58 |
| 41 | 41,840,841 | -- |
| 42 | -- | 42,56,70; 42,144,150 |
| 43 | 43,924,925 | -- |
| 44 | 44,483,485 | 44,240,244 |
| 45 | 45,1012,1013 | 45,60,75; 45,108,117; 45,200,205;45,336,339 |
| 46 | -- | 46,528,530 |
| 47 | 47,1104,1105 | -- |
| 48 | 48,575,577 | 48,64,80; 48,90,102; 48,140,148;48,286,290 |
| 49 | 49,1200,1201 | 49,168,175 |
| 50 | -- | 50,120,130; 50,624,626 |
Now, some of you may be thinking that I have too much time on my hands. Some of you know better. Had I had more time, I could have compiled this sooner. And it might've gone up to 100. And it might have been correct.
Well, I think it's correct, but typos and inadvertant omissions do happen.
Okay, I think I'm done now.
11 comments:
27,120,121 i think should be 17,120,123 becasue that goes into the a, b, b+3 set and the one for line 27 is missing
Thanks.
27,120,123 is correct. All the numbers are multiples of 3, so 121 is obviously wrong.
Proofreading this was a pain, and I'm sure I missed a couple.
Proofreading was also a consideration for stopping at 50.
That, and some of the numbers were getting extremely large.
39-42-65 is wrong
Thanks anyway You almost do my homework for me...
Can't make it too easy for you, can I?
42 is a typo. "Obviously", it should've been 52.
Pythagorean triples can be generated as function of x (x>=3) according to Mir's Generalized Pythagorean triple generation theorem (MGPTGT) which states "let (x, y, z) be a Pythagorean integral triple (base, perpendicular, hypotenuse in that order) then there exist triple(s) for every x (x>=3) because of existence of two integers (j, k) with parity that of x (3 <=j =3
More extensive tables are currently being generated with python.
Dr. (Prof.) Shabir Ahmad Mir, Author
28-45-53 is missing?
28^2 = 784
45^2 =2025
==========
53^2 =2809
Found in a CAASPP practice problem by a talented MS Geometry class...
Thanks for that addition, to which I'll add 36 - 77 - 85.
I haven't thought about this post in years, and it started just from my musings on the subject. I never wanted to write a program to find them all (and I used to be a programmer, so it would be somewhat trivial), and I was doing some exploration of my own into the topic to get my own insights, but I was reluctant to read others' notes on the subject. I realize that this was been done to death by so many others.
Formulas for generating them are out there, and they don't interest me as much. (It becomes a computer problem instead of a math problem at that point.)
My original thinking was much too limited, and had I continued, I would've seen (which I eventually did) that I could search for examples where, say, c = b + 2n^2, where n is an whole number. The fault in my logic was assuming that a should be the smallest number in the triple.
In cases like 3-4-5 or 7-24-25, it's not that c is one more than b, it's that 5 = 3 + 2(1) or 25 = 7 + 2(9).
Also 44 - 117 - 125.
I just ran a short BASIC program to find Triples in the form of a, a+1, c (boring, but I got to practice coding for five minutes), and I got the following:
3 4 5
20 21 29
119 120 169
696 697 985
4059 4060 5741
6437 6438 9104
7422 7423 10497
8815 8816 12467
9800 9801 13860
Thank you for your kind & constructive response. We're happy to help, and have shared this thread with other classes at our school.
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