Will someone give me a LOGIC for this??

All of you may have observe that: 3^2+4^2=5^2 (if not then observe it now)

But instead, if you want to find two numbers such that sum of their squares is equals to square of any specific number then it would be a hard task to find them. But I have found myself a general way to find those numbers.

Let T be the number (say 13) for which you want two numbers m,n such that sum of their squares is equals to T^2 (in our example, 13^2=169). Now divide T by 5 and let the answer be a (13/5=2.6). Now:

m=T-a (m=13-2.6=10.4)

n=T-2a (n=13-2x2.6=7.8)

Here we got our numbers. Now check that (m^2+n^2) will be equals to T^2.

Although it worked for every number I have tried yet but I can't find an explanation for that why one have to divide every number with 5. So please comment any logic you can get for this and also your views about should I get an medal or any award for this discovery.

Note by Vaibhav Jain
4 years ago

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You're basically exploiting the fact that \( (3k)^2 + (4k)^2 = (5k)^2 \). (This is your identity multiplied by \( k^2 \).)

From your formula, \( m = T - a = T - \dfrac{T}{5} = \dfrac{4T}{5} \) and \( n = T - 2a = T - \dfrac{2T}{5} = \dfrac{3T}{5} \)

and therefore \(m^2 + n^2 = \dfrac{4T}{5}^2 + \dfrac{3T}{5}^2 = \dfrac{5T}{5}^2 = T^2 \).

You can do this using other Pythagorean triplets as well, for example, for some number \( S \), let \( b = \dfrac{S}{13} \). and defining \( k = S - b \) and \( l = S - 8b \), we can see that \( k^2 + l^2 = S^2 \)

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However your way is not the only way. Im surprised you found such a specific method. The method I use is basically like an isosceles right triangle. If there is a right triangle that has two legs equal to each other then the hypotenuse is the leg length multiplied by root 2. For example if the there is a right triangle with both legs of side 3 then the hypotenuse is 3(2^0.5) which is 3 times square root of 2. So, if you have only the hypotenuse, you can get two legs easily by dividing the hypotenuse by root 2. So, if in your scenario if the hypotenuse is 13, then two sides can be 13 divided by square root of 2.

Heal Teranaki - 4 years ago

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