Right-angled triangles with integer sides a,b.c,less than 100 such that:c*c=a*a+b*b

Question

$$5*5=3*3+4*4$$

Integers a,b,c
For a=1 to 100
For b=1 to 100
For c=1 to 100
if c*c=a*a+b*b
list a.b,c
Next c
Next b
Next a

I am 74.

I am interested in youth-hood practical mathematics. Child can use strings of 3,4,5 to draw a right-angled triangle. This 3,4,5 triangle also shows a short-cut 5,saves 3+4 by 2. I come across this site by chance. I even forget the Language I used some 50 years ago. I would be very much obliged to be enlighted on this question.

Answer 1

You are asking for what are called "Pythagorean triples": sets of integers satisfying $a^2+b^2=c^2$. There is a formula for generating all such sets of integers, and you can read all about it in the Wikipedia article: Pythagorean triple.

Near the top of that article is a list which is almost precisely what you are looking for. The only difference is this: if we have a triple such as $(3,4,5)$, you can obtain others by multiplying all three numbers by the same factor, i.e., we have $(6,8,10)$, $(9,12,15)$, etc. These multiples are not included in the list at Wikipedia (they are not what we call "primitive"), but you can find them all without too much work.