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Difference between revisions of "2003 IMO Problems/Problem 6"

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== Problem ==
 
== Problem ==
 
Let <math>p</math> be a prime number. Prove that there exists a prime number <math>q</math> such that for every integer <math>n</math>, the number <math>n^p-p</math> is not divisible by <math>q</math>.
 
Let <math>p</math> be a prime number. Prove that there exists a prime number <math>q</math> such that for every integer <math>n</math>, the number <math>n^p-p</math> is not divisible by <math>q</math>.
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== Solution ==
 
== Solution ==

Latest revision as of 08:39, 5 July 2024

2003 IMO Problems/Problem 6

Problem

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Let N be $1 + p + p^2 + ... + p^{p-1}$ which equals $\frac{p^p-1}{p-1}$ $N\equiv{p+1}\pmod{p^2}$ Which means there exists q which is a prime factor of n that doesn't satisfy $q\equiv{1}\pmod{p^2}$. \\unfinished

See Also

2003 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
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