Knots and Pythagorean Triangles

Note that we can represent any Pythagorean triplet as $(m(u^2 - v^2), 2muv, m(u^2 + v^2))$ for some integers $u > v > 0; m > 0$ and that $k$ is just the sum of the sides (assuming that the distance between two consecutive knots is 1).

Also, notice $(6,8,10)$ gives $k = 24$ .

We prove that for $k < 24$ , no right triangle exists. This is same as showing that the following equation has no integer solutions :

$k = 2mu^2 + 2muv$

We see that no solutions exist if $k$ is odd. This narrows it down to $k = 14, 16, 18, 20, 22$ .

Now, put $k = 2n$ so that the equation now becomes the following :

$n = mu(u+v)$

Note that there is no solution for $n$ being of the form $p, p^2$ or $p^3$ for some prime $p$ .

This further narrows down the search to $n = 10$ . Exhaustion now shows that no solution exists.

i'll show a very simple answer:

consider the distance between one knot as 1 unit . now to proceed with the solution .we must find the next Pythagorean triple . since the current triplet is (3,4,5) the next triplet is (6,8,10) and hence the answer is $6+8+10=24$

We have to remember Phytagorean to solve this question. We know that figure-2 has 12 knots. This is one of triplet phytagorean, which if we add all the sides equal to 12. Yes, the number is (3,4,5). Then we should find the minimum k > 12. For this case, we only need to add the next triplet phytagorean. The next triplet phytagorean ( after 3,4,5) is (6,8,10) Therefore (6 + 8 + 10) = 24. So, the answer is 24