On Solutions of the Diophantine Equations $p^4 + q^4 = z^2$ and $p^4-q^4= z^2$ when p and q are Primes

Abstract

. In this article, we consider the equations p 4 + q 4 = z 2 and p 4 - q 4 = z 2 when p , q are primes and z is a positive integer. We establish that p 4 + q 4 = z 2 has no solutions for all primes 2 ≤ p < q . For p 4 - q 4 = z 2 when 2 ≤ q < p , it is shown: (i) For q = 2 the equation has no solutions. (ii) The equation z 2 = p 4 – q 4 = ( p 2 – q 2 )( p 2 + q 2 ) is impossible when each factor is equal to a square. (iii) When each factor is not a square, conditions which must be satisfied simultaneously are determined for p 2 and q 2 . For all primes p < 2100, these conditions are not fulfilled simultaneously. It is conjectured for all primes p > 2100 and q > 2 that the equation has no solutions.