Select Page

# 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

## Tetrahedron Problem | AIME I, 1992 | Question 6

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Tetrahedron Problem.

## Triangle and integers | AIME I, 1995 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Triangle and integers.

## Functional Equation Problem from SMO, 2018 – Question 35

Try this problem from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation. You may use sequential hints if required.

## Sequence and greatest integer | AIME I, 2000 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Sequence and the greatest integer.

## Arithmetic sequence | AMC 10A, 2015 | Problem 7

Try this beautiful problem from Algebra: Arithmetic sequence from AMC 10A, 2015, Problem. You may use sequential hints to solve the problem.

## Series and sum | AIME I, 1999 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Series and sum.

## Inscribed circle and perimeter | AIME I, 1999 | Question 12

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.

## Problem based on Cylinder | AMC 10A, 2015 | Question 9

Try this beautiful problem from Mensuration: Problem based on Cylinder from AMC 10A, 2015. You may use sequential hints to solve the problem.

## Cubic Equation | AMC-10A, 2010 | Problem 21

Try this beautiful problem from Algebra, based on the Cubic Equation problem from AMC-10A, 2010. You may use sequential hints to solve the problem.

## Median of numbers | AMC-10A, 2020 | Problem 11

Try this beautiful problem from Geometry based on Median of numbers from AMC 10A, 2020. You may use sequential hints to solve the problem.