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

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

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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

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 == Problem ==
-p is a prime number. Prove that for every p there exists a q for every positive integer n, so that <math>n^p-p</math> can't be divided by q.
+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>.
 == Solution ==

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

Latest revision as of 08:39, 5 July 2024

Contents

2003 IMO Problems/Problem 6

Problem

Solution

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