The author gives elementary proofs of the following theorems. ( 1 ) If 2|mn, q =pt2 + 2, then the equation qm= 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. |