Square numbers and square roots?

If n is a positive integer: 2n+1 will be an odd number

(2n+1), ((2n+1)²-1)/2 and ((2n+1)²+1)/2 will always be a Pythagorean triple

Examples:
n=1: 3, 4, 5
n=2: 5, 12, 13
n=3: 7, 24, 25
n=4: 9, 40, 41

This, however, is NOT the entire set of Pythagorean triples

Yes, there are. These sets of numbers are called "Pythagorean triples", since they are integer solutions to Pythagoras's theorem, a² = b² + c².

There are infinitely many of them... 3² + 4² = 5² is the common example, being the smallest... 5² + 12² = 13² is another.

Sure. What about

3² + 4² = 25

√(25) = 5

√200=10√2 √32=4√2 √200-√32=10√2-4√2=6√2

Yes.

3^2 + 4^2 = 5^2
5^2 + 12^2 = 13^2
7^2 + 24^2 = 25^2
9^2 + 40^2 = 41^2
11^2 + 60^2 = 61^2
13^2 + 84^2 = 85^2
... ...
(can you find the relationship between them?)

3 and 4:
3^2 = 9
4^2 = 16
9 + 16 = 25
sqrt25 = 5