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.