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[SOLVED] Polynomial with integer coefficients

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,812
Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.
 

kaliprasad

Well-known member
Mar 31, 2013
1,334
The polynomial is P(x)

we have m-n divides P(m) - p(n)

Let the given condition is true

So $a-b | P(a) - p(b)$

or $a-b | b- c$

Similarly

$b - c | c- a$

And $ c- a | a - b$

From above 3 have

$a-b | b-c | c- a | a-b$


So all are same and hence a = b= c which is a contradiction.


so condition can not be satisfied simultaneously