Fun with Mathematics-5

 Fun with Mathematics-3

     Fermat Last Theorem

Fermat’s finding

Pierre de Fermat (1601-1665), the French mathematician came up with one of the most famous theorems of all time. It is still being a puzzle!

Equation of the form a^n + b^n = c^n, has no solutions, when n is a whole number greater than 2 and when a,b,c are all positive whole numbers.

Special cases

When n = 1, a + b = c

One can show very easily the sum of any two numbers is equal to a number.

If a + b = c, then a^2 + b^2 ≠ c^2

If a+b = c then (a+b)^2 = axa + bxb +2 axb = c^2

Or a^2 + b^2  < c^2

Similarly,

If a+b = c, then a^3 + b^3 ≠ c^3

If a+ b = c, then (a+b)^3 = a^3 + b^3 + 3ab(a+b) = c^3

Or a^3 + b^3 < c^3

In general, if a+b = c then  a^n + b^n   > c^n  for n <1

                                           a^n + b^n = 1 for n = 1

                                           a^n +b^n   < c^n  for n ≥ 1

When n = 2, a^2 + b^2 = c^2

The set of three numbers (a,b,c) is called Pythagorean triple. It is not so easy to construct

such relations as that of a+b = c. Only a certain set of values of a,b,c satisfies the

Pythagorean rlation,a^2 + b^2 = c^2.

If a^2 + b^2 = c^2 , then (a+b)^2 =a^2 + b^2 + 2ab = c^2 + 2ab

Or (a+b) = c √ [1 + (2ab)/c^2]

It means that a+b > c

Let us suppose that a+b = c+ d, where d is a small positive number added to c to balance the relation. 

(a+b)^2 = (c+d)^2

 a^2 + b^2 + 2ab = c^2 + d^2 + 2cd

a^2 + b^2 – c^2 = d^2 – 2(ab-cd)

since a^2 + b^2 = c^2, we have d^2 = 2(ab-cd) must be true.

(a+b)^3 = (c+d)^3

a^3 + b^3 + 3ab(a+b) = c^3 + d^3 + 3cd(c+d)

a^3+b^3 – c^3 = d^3 – 3(a+b) [ab-cd]

                         = d^3 – [3(a+b)d^2]/2

a^3+b^3-c^3-d^3 is negative or c^3 + d^3  > a^3 + b^3

If a^3 + b^3 = c^3 to be true, d^3 – (3/2)(a+b) d^2 = 0

Or d = (3/2) (a+b) = (3/2)(c+d) which is not possible

a^3 + b^3 – c^3 = d^3 – (3/2)(c+d) d^2

         = d^3 – 3/2cd^2 – 3/2 d^3

         = - 3/2 cd^2 – ½ d^3

      = - d^2/2 (3c+d)

or c^3 > a^3 + b^3

Since it is a negative quantity a^3 + b^3 < c^3

If a^3 + b^3 = c^3 exists, d= 3(a+b)/2= 3/2© + 3/2(d) which is absurd

If a^2 + b^2 = c^2 ,then a^n + b^n  > c^n  for n < 2

                                       a^n + b^n = c^n  for n = 2

                                       a^n + b^n  < c^n for n > 2

 Fermat’s last theorem has tantalized mathematicians world over for more than 350 years

In fact Fermat had scribbled the proof for the theorem in the margin of a book adding that although the proof was excellent, the space available was inadequate to hold it. In 1994

Andrew Wiles of Princeton University and Richard Taylor of Cambridge University provided a proof for it. In this series of articles we will try to find some potential ways to tackle out the problem in a different angle of approach.