Fun with mathematics-12

Fermat’s last theorem - 3

Proof.5.

It seems very easy to prove that a^4 +b^4 = c^4 is not existing.

All the three root numbers a,b and c cannot be even or odd. If they

are even, it will be reducible until one of the roots becomes odd.

As the sum of two odd numbers cannot result with odd, all of them

cannot be odd.

Fourth power of an odd number is always in the form of 16n +l,

while that of the even number is in the form of 16n.

If a and b are odd, then c should necessarily be even. If their

fourth powers are expressed in multiples of 16,

{16[n(a)]+1} + {16[n(b)]+1} = 16[n(c)]

16[n(a)+n(b)]+2 = 16[n(c)]

n(c) cannot be a fraction. Hence the possible existence of the

relation is rooted out.

 

In general if a^n + b^n = c^n is existing, then it can be

expressed as

x^n + y^n = 1, where x = a/c and y = b/c.

Since c>a, and c>b,both x and y will fractions. Such

numbers are called rational numbers.

For n >2, x^n +y^n =1 has no non-zero rational

solutions, though it has infinite number of non-zero

solutions which are not rational.