Tag Archives: prime

2013-08 Minimum of a set involving polynomials with integer coefficients

Let \( p \) be a prime number. Let \( S_p \) be the set of all positive integers \( n \) satisfying
\[
x^n – 1 = (x^p – x + 1) f(x) + p g(x)
\]
for some polynomials \( f \) and \( g \) with integer coefficients. Find all \( p \) for which \( p^p -1 \) is the minimum of \( S_p \).

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2008-1 Distinct primes (9/4)

Let \(n\) be a positive integer. Let \(a_1,a_2,\ldots,a_k\) be distinct integers larger than \(n^{n-1}\) such that \(|a_i-a_j|<n\) for all \(i,j\).

Prove that the number of primes dividing \(a_1a_2\cdots a_k\) is at least \(k\).

\(n\)은 양의 정수라 하자. \(n^{n-1}\)보다 큰 \(k\)개의 서로 다른 정수 \(a_1,a_2,\ldots,a_k\)가 모든 \(i,j\)에 대해서 \(|a_i-a_j|<n\)을 만족한다고 하자.

이때 \(a_1a_2\cdots a_k\)의 약수인 소수의 개수는 \(k\)개 이상임을 보여라.

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