User:Bhowmickr

$a^n-b^n=c^2-d^2$ like $7^3-4^3=48^2-45^2$

$x^n=z^2-y^2$ like $3^3=14^2-13^2$ ,

$a^n-b^n=(a-b)^n+nk(a-b)$,here when "a" is odd natural number and "b" is any natural number and "n" is any prime or its factors are the same prime(here the multiples of 5 like 25,125 etc. are excluded but the case of 5 is included) then "k" will always be a natural number.

From this it can be proved very easily and in an elementary way that it is impossible to write

$a^n-b^n=x^n$

or

$a^n=x^n+b^n$

for n>2.