On a Problem of Hall

Abstract:

      The author gives elementary proofs of the following theorems. ( 1 ) If 2|mn, q =pt² + 2, then the equation q^m= pⁿ+ 2, p, q are primes, m> 1, n> 1, (*)has no solution.( 2 ) The equation (p+2)^2m₁=pⁿ+ 2 has no solution with m₁> 1, n>1 . (3) If n = 2m₁, q =pt² + 2, then the equation (*) has no solution. Clearly, if q=p + 2, then the equation ( *) has no solution.