Find a second-degree polynomial with integer coefficients, p(x) = ax² + bx + c, such that p(1), p(2), p(3), and p(4) are perfect squares (that is, squares of integers), but p(5) is not.

Evaluate for x=1,2,3,4 and set equal to A^2,B^2,C^2,D^2. Then eliminating a,b,c gives A^2 + 3C^2 = D^2 + 3B^2 so what's needed are integers expressible as the sum of a square and 3 times a square in two ways.

Posted by xdog on 2013-08-16 11:54:48