## Saturday, March 12, 2011

### Pythagorean Triples: Rides Again!

Yes, my friends and relatives think I'm crazy, but this is what I do....

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):
33, 56, 65 and 36, 77, 85

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

Anonymous said...

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
52 + 122 = 25 + 144 = 169
132 = 169

Kartik Kwatra said...

hi Mrbuker, how do u explain triple : 39,80,89 ?

(x, why?) said...

Thanks for writing. (It's Mr. Burke, by the way.)

I 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.

Kartik Kwatra said...

Thanks Mr. Burke.