Tag Archives: prime

2019-11 Smallest prime

Find the smallest prime number \( p \geq 5 \) such that there exist no integer coefficient polynomials \( f \) and \( g \) satisfying
p | ( 2^{f(n)} + 3^{g(n)})
for all positive integers \( n \).

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2008-6 Many primes

Let f(x) be a polynomial with integer coefficients. Prove that if f(x) is not constant, then there are infinitely many primes p such that \(f(x)\equiv 0\pmod p\) has a solution x.

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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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