Since A<N, we set A+M=N, squaring gives A²+2AM+M²=N²
thus B=2AM+M², and since B | 2A², we can set BK=2A², ie
Thus 2 | M²K. Now assume that K isnt divisible by 2. Then 2 | M, and we can set M=2m, which gives.
now we see that 2 | A, and we set A=2a, which gives
but this is the same form as equation (1), thus this process will continue forever, and we have a contradiction. Thus 2 | K, and in (1) we set K=2k, which yields:
adding M²k² to both sides:
thus k(k+1) must be a square, but the product of two consecutive positive integers cant be a square.
Edited on May 24, 2008, 3:53 pm