Find all pairs of prime numbers \( (p, q) \) such that \( pq \) divides \( p^p + q^q + 1 \).

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Find all pairs of prime numbers \( (p, q) \) such that \( pq \) divides \( p^p + q^q + 1 \).

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 \( n \).

Let \( p_n \) be the \(n\)-th prime number, \( p_1 = 2, p_2 = 3, p_3 = 5, \dots \). Prove that the following series converges:

\[

\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.

\]

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 \( S_p \).

Prove that there is a constant c>1 such that if \(n>c^k\) for positive integers n and k, then the number of distinct prime factors of \(n \choose k\) is at least k.

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.

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.