Proving no polynomial $P(x)$ exists such that $P(a) = b$, $P(b) = c$, $P(c) = a$

Question

If $P(x)$ is a polynomial with integer coefficients and $a, b ,c$ are three distinct integers, then show that it is impossible to have $P(a) = b$, $P(b) = c$, $P(c) = a$.

Have you tried seeing what happens under Lagrange interpolation? — Just_a_fool, May 23 '14 at 07:11
http://math.stackexchange.com/questions/387721/polynomial-pa-b-pb-c-pc-a — user263326, Aug 04 '16 at 20:44

score 2 · Accepted Answer · answered May 23 '14 at 07:17

2

Hint: $b-c$ is divisible by $a-b$, and ...

answered May 23 '14 at 07:17

Robert Israel

470,583

I got your point but what next – Achal Kumar May 23 '14 at 07:25
... and $a-b$ is divisible by $c-a$, and ... – Robert Israel May 23 '14 at 07:29
This is an old USAMO, I think. – chubakueno May 24 '14 at 17:37

Proving no polynomial $P(x)$ exists such that $P(a) = b$, $P(b) = c$, $P(c) = a$

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