 # Understand the problem

For a polynomial $f(x)$ with integer coefficients and degree no less than $1$, prove that there are infinitely many primes $p$ which satisfies the following. There exists an integer $n$ such that $f(n) \not= 0$ and $|f(n)|$ is a multiple of $p$.
##### Source of the problem
Korea Junior MO Problem 7
Number Theory
8/10
##### Suggested Book
Elementary Number Theory by David Burton

Do you really need a hint? Try it first!

Well, remember the proof that the set of prime numbers is infinite? We started with the assumption that let there be a finite number of prime numbers and then reached a contradiction that there needs to be another extra prime number given that set. Hence, the set of prime numbers is infinite. This problem is also famously known as Schur’s Theorem. Observe that the problem can be restated as every nonconstant polynomial p(x) with integer coefficients if S is the set of all nonzero values, then the set of primes that divide some member of S is infinite. Let us start by assuming that the set is indeed finite. Let $A$ this set of primes $p$ such that $\exists n$ such than $f(n)\ne 0$ and $p|f(n)$. Let |A| be finite.

If $f(0)=0$ the result is immediate since $p|f(p^n)$ $\forall p$ (just choose $n$ such that $f(p^n)\ne 0$ and so any prime $p\in A$. Now let’s take the case when f(0) is non-zero. Let’s take $f(x) = a_n.x^n + … a_1.x + f(0)$.  Now, $f(c.f(0)) = a_n.{c.f(0)}^n + … a_1.f(0) + f(0) = f(0).( a_n.c.{cf(0)}^{n-1} + … + a_2.c^2.f(0) + a_1.c + 1 )$. Can you give some appropiate  c to show that another prime must exist?
Take c = product of all the primes in A.  Prove that it implies some other prime must exist which is not in A.
Now, $f(c.f(0)) = a_n.{c.f(0)}^n + … a_1.f(0) + f(0) = f(0).( a_n.c.{cf(0)}^{n-1} + … + a_2.c^2.f(0) + a_1.c + 1 )$. Observe that if we take c as mentioned then, i.e. c = product of all the primes in A. Then all f(c.f(0)) must be coprime to all the primes in A. Therefore, it must have a prime factor other than those in A. Hence, a contradiction in the finiteness in A. QED.

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

## Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

## Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

## Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

## Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

## Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.

## Rectangular Piece of Paper | AMC 10A, 2014| Problem No 22

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2014. Problem-23. You may use sequential hints to solve the problem.

## Probability in Marbles | AMC 10A, 2010| Problem No 23

Try this beautiful Problem on Probability from AMC 10A, 2010. Problem-23. You may use sequential hints to solve the problem.

## Points on a circle | AMC 10A, 2010| Problem No 22

Try this beautiful Problem on Number theory based on Triangle and Circle from AMC 10A, 2010. Problem-22. You may use sequential hints to solve the problem.

## Circle and Equilateral Triangle | AMC 10A, 2017| Problem No 22

Try this beautiful Problem on Triangle and Circle from AMC 10A, 2017. Problem-22. You may use sequential hints to solve the problem.