p(x) is a polynomial with integral coefficients. Show that there are no solutions to the equations p(a) = b, p(b) = c, p(c) = a, with a, b, c distinct integers.
Solution
Suppose there a, b, c satisfy the equations. Then p(x) = (x - a)q(x) + b = (x - b)r(x) + c = (x - c)s(x) + a for some polynomials q(x), r(x), s(x) with integer coefficients. Hence (b - a)q(b) + b = p(b) = c, so (b - a) divides (c - b). Similarly, (c - b) divides (a - c), and (a - c) divides (b - a). But (b - a) divides (c - b) divides (a - c) implies that (b - a) divides (a - c). So we have (b - a) and (a - c) dividing each other. Hence (b - a) = ±(a - c).
If b - a = a - c, then b - c = (b - a) + (a - c) = 2(a - c). But (b - a) divides 2(a - c) (and both are non-zero since a, b, c are distinct), so that is impossible. If b - a = -(a - c), the b = c, contradicting the fact that they are distinct. So there are no solutions.
© John Scholes
jscholes@kalva.demon.co.uk
19 Aug 2002