A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if - $p$ is an odd prime number, - $a, b$, and $c$ are distinct and - $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$. a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.

Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

Let $m,n$ be positive integers and $p$ be a prime number. Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.

$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$.

Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.

Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.

Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer. Show that $p = q$.

Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.

Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime. [i]Proposed by Jack Gurev[/i]

We call a positive integer [i] good[/i] if the sum of the reciprocals of all its natural divisors are integers. Prove that if $ m $ is a [i]good [/i] number, and $ p> m $ is a prime number, then $ pm $ is not [i]good[/i].

Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$ Show that $p=q$.

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Two positive integers $n,m\ge 2$ are called [i]allies[/i] if when written as a product of primes (not necessarily different): $n=p_1p_2...p_s$ and $m=q_1q_2...q_t$, turns out that: $$p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t$$ (a) Show that the biggest ally of any positive integer has to have only $2$ and $3$ in its prime factorization. (b) Find the biggest number which is allied of $2021$ .

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

Let the sequence ($a_n$) be defined by $a_n = n^6 +5n^4 -12n^2 -36, n \ge 2$. (a) Prove that any prime number divides some term in this sequence. (b) Prove that there is a positive integer not dividing any term in the sequence. (c) Determine the least $n \ge 2$ for which $1989 | a_n$.

a) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31$ (the product of primes $2$ to $31$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$. b) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$ (the product of primes $2$ to $37$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.