MHB Polynomial with integer coefficients

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The discussion centers on proving that for three distinct integers \(a\), \(b\), and \(c\), a polynomial \(P\) with integer coefficients cannot satisfy the conditions \(P(a)=b\), \(P(b)=c\), and \(P(c)=a\) simultaneously. The key argument involves analyzing the implications of these conditions on the polynomial's behavior and the properties of integers. It is established that such a polynomial would lead to contradictions regarding the distinctness of \(a\), \(b\), and \(c\). Therefore, the conclusion is that no polynomial with integer coefficients can meet all three conditions at once. This result highlights the limitations of polynomial mappings in relation to distinct integers.
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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.
 
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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
 

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