Polynomial problem involving divisibility, prime numbers, monotonyLet f be a polynomial function, with integer coefficients, strictly increasing on N such that f ( 0 ) = 1. Show that it doesn't exist any arithmetic progression of natural numbers with ratio r &gt; 0 such that the value of function f in every term of the progression is a prime number. I believe that the solution includes a reductio ad absurdum, but I don't know how to solve it.

Question

Polynomial problem involving divisibility, prime numbers, monotonyLet f be a polynomial function, with integer coefficients, strictly increasing on       N   such that   f  (  0  )  =  1. Show that it doesn&#039;t exist any arithmetic progression of natural numbers with ratio   r  &amp;gt;  0 such that the value of function f in every term of the progression is a prime number. I believe that the solution includes a reductio ad absurdum, but I don&#039;t know how to solve it.

cenjene9gw · Accepted Answer

Explanation:If progression is   a  +  n  b, and   f  (  a  )  =  p, then   f  (  a  +  p  n  b  )  ≡  0    (  mod    p  ) and absolute value is more then p for large n.