Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$.

Question

Let $f(x)$ be a polynomial in $x$ With integer coefficient. If for natural numbers $a,b,c$, $f(a)=b, f(b)=c, f(c)=a$ Prove that $a=b=c$.

It seems likely that we are expected to assume $f$ has integer coefficients. — André Nicolas, Jul 03 '15 at 15:58

score 9 · Answer 1 · edited Apr 13 '17 at 12:19

9

This is not true: Consider the polynomial $f(x)=-\frac{3 x^2}{2}+\frac{11x}{2}-2$. Then $f(1)=2$, $f(2)=3$, $f(3)=1$.

(Edit: This is the answer to a previous version of the question, which didn't specify that the coefficients must be integers. For a hint or an answer to the modified version, see André Nicolas' answer above or this question.)

edited Apr 13 '17 at 12:19

Community

1

answered Jul 03 '15 at 15:52

n55

1,563

1

The question has been modified, but your answer to the previous version is still valuable. It may be a good idea to modify the answer to make clear what it answers. – André Nicolas Jul 03 '15 at 16:19
Done. Thanks for the suggestion! – n55 Jul 03 '15 at 16:26

score 4 · Accepted Answer · answered Jul 03 '15 at 16:02

4

Hint: For the modified question, note that $a-b$ divides $f(a)-f(b)$.

So $a-b$ divides $b-c$. Similarly, $b-c$ divides $c-a$ and $c-a$ divides $a-b$.

answered Jul 03 '15 at 16:02

André Nicolas

507,029

Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$.

2 Answers2