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.