Squares from products of integers
Banks, William David, 1964-
Van Der Poorten, A. J.
Notice that 1_2_3_4+1 = 52 , 2_3_4_5+1 = 112 , 3_4_5_6+1 = 192 , . . . . Indeed, it is well known that the product of any four consecutive integers always differs by one from a perfect square. However, a little experimentation readily leads one to guess that there is no integer n, other than four, so that the product of any n consecutive integers always differs from a perfect square by some fixed integer c = c(n) depending only on n. The two issues that are present here can be readily dealt with. The apparently special status of the number four arises from the fact that any quadratic polynomial can be completed by a constant to become the square of a polynomial. Second,  provides an elegant proof that there is in fact no integer n larger than four with the property stated above. In  one finds a reminder that a polynomial taking too many square values must be the square of a polynomial (see [4, Chapter VIII.114 and .190], and ). One might therefore ask whether there are polynomials other than integer multiples of x(x + 1)(x + 2)(x + 3) and 4x(x + 1), with integer zeros and differing by a nonzero constant from the square of a polynomial. We will show that this is quite a good question in that it has a nontrivial answer, inter alia giving new insight into the results of .
Mathematics publications (MU)