MHB Polynomial challenge: Show that not all the coefficients of f(x) are integers.

[castor28]

$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$.

Show that not all the coefficients of $f(x)$ are integers.

 

[kaliprasad]

$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$.

Show that not all the coefficients of $f(x)$ are integers.

Let all coefficients be integers

Spoiler

we have m-n divides $f(m)-f(n)$
so p- q | f(p) - f(q) | q-r

similarly
q -r | r-p
and r-p | p - q

from above as p - q | q-r | r- p | p- q (all nteger multiples) so all are same and hence a contradiction

so all coefficients cannot be integers

 

[castor28]

Let all coefficients be integers

Spoiler

we have m-n divides $f(m)-f(n)$
so p- q | f(p) - f(q) | q-r

similarly
q -r | r-p
and r-q | p - q

from above as p - q | q-r | r- p | p- q (all nteger multiples) so all are same and hence a contradiction

so all coefficients cannot be integers

That is quite correct (except for a small typo: $r-q\mid p-q$ should be $r-p\mid p-q$).
Congratulations!:)

 

[kaliprasad]

That is quite correct (except for a small typo: $r-q\mid p-q$ should be $r-p\mid p-q$).
Congratulations!:)

Thanks castor. Corrected the same inline for the flow.