State a general law suggested

Question

Observe that

$3^2+4^2=5^2$

$5^2+12^2=13^2$

$7^2+24^2=25^2$

$9^2+40^2=41^2$

State a general law suggested by these examples, and prove it.

For $ a ^ 2 = b ^ 2 + c ^ 2, and (a, b, c) = 1 $. I was thinking of the early Pythagorean triples

Answer 1

Starting with the smallest you may see

• $1\cdot 3 + 1 = 4$
• $2\cdot 5 + 2 = 12$

• $3\cdot 7 + 3 = 24$ ...

• Guess: $(2k+1)^2 + (k(2k+1)+k)^2 = (k(2k+1)+k + 1)^2$ for $k \in \mathbb{N}$ or simplified

• $(2k+1)^2 + (2k(k+1))^2 = (2k(k+1) + 1)^2$ ... which can be verified by direct calculation

Answer 2

We need $$(n+1)^2-n^2=2n+1$$

to be perfect square

As $2n+1$ is odd,let $2n+1=(2m+1)^2$ where $m$ is any integer

$\iff n=2m(m+1)$