Recently, in the comments to my post Pythagorean Triples: An Easier Way, blogger Keith issued a friendly challenge involving primitive triples with the same hypotenuse.
I have to be honest here: it never occurred to me that two primitive triples would have the same hypotenuse for two reasons: first, I hadn't really looked at numbers that went that high (and I'm certainly not using them in class); second, they didn't fit my three models for Pythagorean Triples: a, b, b+1; a, b, b+2; and a, a+1, c.
I explained why b+3 didn't work, but I never pondered if b+9 or b+18 would work. And, I know now, it would have.
Basically, I wanted to investigate this myself, just for fun. So I didn't use the formulas I knew about, namely pick an m and n and calculate a=m2-n2, b=2mn, and c=m2+n2.
That will give you every triple there is, primitive or not, with lots of repeats, in a very disorganized manner. (For one thing, b will always be the even number, not the middle number.)
But since I didn't use it, and despite the graphic I generated in this comic, I omitted the following triples from my original list of Pythagorean Triples (3-50):
This, of course, got me to wondering why some hypotenuses would have more than one. Well, that's kind of obvious, depending on whose lists of numbers you look at. But, of course, I'm more interested in seeing if there's any pattern to be found.
In the meantime, here's an updated list:
Pythagorean triples, sorted by the shortest side, from 3 to 50.
| 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 | 33,56,65; 33,544,545 | 33,44,55; 33,180,183 |
| 34 | -- | 34,288,290 |
| 35 | 35,612,613 | 35,84,91; 35,120,125 |
| 36 | 36,77,85; 36,323,325 | 36,48,60; 36,160,164; 36,105,111 |
| 37 | 37,684,685 | -- |
| 38 | -- | 38,360,362 |
| 39 | 39,760,761 | 39,42,65; 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 |
The explanation is good and I am here to share my views about Pythagorean triple-Pythagorean triple is a set of three non-zero numbers in which the sum of the squares of two numbers is equal to the square of the third number. To explain this I am giving an example- Set (5, 12, 13) is a Pythagorean triple,when
ReplyDelete52 + 122 = 25 + 144 = 169
132 = 169
hi Mrbuker, how do u explain triple : 39,80,89 ?
ReplyDeleteThanks for writing. (It's Mr. Burke, by the way.)
ReplyDeleteI explain 39, 80, 89 by saying it was generated by a Excel spreadsheet.
This was my own little exploration, without resorting to using formulas for generating triples, which I've seen in the past.
The one problem with my own trials is that they didn't go high enough to see a trend, which I later noticed. However, I haven't gotten around to writing an addendum.
The connection isn't between the two biggest numbers, but with the two odd numbers. If you look at the primitive triples, the difference between two odd numbers will be in the form of 2n^2.
So in cases when c=b+2, that's actually 2 * (1)^2.
In the cases where c=b+1, I wasn't looking at the fact that c=a+2n^2, where n is a whole number.
In this example, 89 - 39 = 50, which is 2*(5)^2.
When I have time, I'll have to write up an addendum. Thanks for writing.
Thanks Mr. Burke.
ReplyDeleteActually I had already submitted the comment when i realized that I had mis-typed your name. Sorry for that.
So now you have brought a new twist.
I had already devised 5 points to find out triples. b+1 , b+2 , b+9,b+18 and a+b/c were the ones, and from the GMAT and CAT perspective ( which are the tests im preparing for) these 5 points would be sufficient enough to find large enough triple ( somewhere in the range of 200 to 300). And i'm afraid there are not any primitive triples below this range which could escape these 5 points.(need an answer for that :) )
So now do I have to rethink on the method of calculation using the latest twist or my methods would suffice?