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PYTHAGORAS AND 300 YEARS CHALLENGE




      Everybody knows Pythagoras theorem. The figure 1 states it clearly.
If sides are equal to unity, what will happen to hypotenuse
c^2=1^2+1^2 = 1+1 = 2
c = root of 2  = 1.414... (figure 2)
Hypotenuse turns into root of 2 , a never ending irrational number.

Suppose hypotenuse is 1, what will be the Pythagoras relationship?
According to trigonometry, adjacent side is 1*cos x and opposite side is 1*sin x.  Applying Pythagoras theorem we get, 

 (1*cos x)^2+(1*sin x.)^2 = 1^2
(cosx)^2+(sinx)^2=1
So this is the trigonometric equivalent of Pythagoras equation.(figure 3)

 We can extend the Pythagoras principle to higher dimensions.  For example, we can calculate the diagonal inside the 3 D cube as shown in the figure 4.



We can link number of triangles and apply the Pythagoras theorem as shown in the figure 5
.  
Pythagoras equation is satisfied by some whole numbers or integers as given below.

 x^2+y^2 = z^2

 3^2+4^2 = 5^2
5^2+12^2 = 13^2
7^2+24^2 = 13^2

 The set of 3 integers(3,4,5) are called Pythagorean triples or triplets.  There are infinity number of Pythagorean triplets.

 We will slightly change the equation. That is, instead of square we will put cube.

 x^3+y^3 = z^3

 Does this equation have any whole number solution?  People were not able to find a single triplet for a long time..  A mathematician by name pierre de fermat said "there is no integer solution at all for this equation".  He further asserted that 

 x^n+y^n = z^n

 has no whole number solutions for n greater than 2.  He claimed even if all the mathematicians in the world spent eternity looking for a solution to the equation.  They would fail to find one.  This is known as fermat's last theorem.
This theorem challenged mathematical community for the last 300 years.  Recently, Mr. Andre Weil proved fermat's last theorem in 1995 and claimed the prize money of 100000 marks.

 Science update:  German scientists are testing an artificial sun, a massive light array for making hydrogen fuel by dissociating water molecules.   
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