[NFB-Science] some fun facts about right triangles

[John Miller]

Hello,
Do you have some tips or tricks you would like to share about STEM?

I have found that I sometimes learn information easier surfing the web on my iPhone than I do using Microsoft Edge with Windows.
The other day I was surfing the web about the Pythagorean theorem.
I came across the definition of Pythagorean triples.
Basically, these are a set of three positive integers whose lengths correspond to the sides of a right triangle.

When I find a website on the iPhone that I like, I click on share, then mail, and then mail the link to myself.
I can view the link in more detail with Microsoft Edge at a later time or go back to it when I like from the copy in my e-mail.

One right triangle you might be quite familiar with has edges with the lengths 3, 4, and 5.
One website mentioned a Pythagorean triple 9, 40, 41.
It caught my attention that two of the members of the Pythagorean triple only differed by one.
Some quick mental math proves this is a triple.
41^2 = (40+1)^2 = 40^2 + 40 + 40 + 1 = 1600 + 80 + 1 = 1681
and the sum of 9^2 and 40^2 = 81 + 1600 also equals 1681.

I got to wondering just what are the list of all Pythagorean triples whose sides are no more than one hundred.
Are there a lot of them or not so many?

I wrote software to generate the list.
I am happy to share the software written in Matlab with anyone who would like it.
I encourage my blind fellow scientists to write software to answer questions that they might have in STEM.
It can open professional opportunities in STEM and getting the answer can be quite rewarding.

Here is the list of all Pythagorean triples with two elements of the set differing by one and all elements no more than one hundred:

a, b, c
3, 4, 5
5, 12, 13
7, 24, 25
9, 40, 41
11, 60, 61
13, 84, 85
20, 21, 29

You can see that the 9, 40, 41 entry is the fourth in the list.

Here are the two related websites I came across:
3 4 5 triangles definition:
https://na01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.mathopenref.com%2Ftriangle345.html&data=05%7C01%7C%7C85d64247385f4948977308db0153baaf%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C638105231347402968%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=jb%2FlejYHQQQe2FqFfb8KVJZdaBrtt9J5ICeGrJLgVI8%3D&reserved=0

Pythagorean triples definition:
https://na01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.cuemath.com%2Fgeometry%2Fpythagorean-triples%2F&data=05%7C01%7C%7C672710c08baf4bf4d51b08db01556f84%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C638105238678116206%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=ayYs8S%2Flq7tr37hUTdToGmr9AZ0LnEqeqtWPKuqMRM0%3D&reserved=0

Very best,
John