Solution to puzzle 95: Integer polynomial
Let P be a polynomial with integer coefficients. If a, b, c are distinct integers, show that
- P(a) = b,
- P(b) = c,
- P(c) = a,
cannot be satisfied simultaneously.
Suppose, without loss of generality, a < b < c.
By the polynomial remainder theorem, P(c) − P(a) = (c − a)Q(c), for some polynomial Q, with integer coefficients.
Hence c − a divides a − b.
But this is impossible, as |c − a| > |a − b|, and a 
b.
Therefore the above system of equations cannot be satisfied simultaneously.
Source: Traditional