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 $p = q$.

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

Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.

Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p $ is divisible by $p$ for all integers $x.$

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 $0$.

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

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.

For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?

Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$ (Paolo Leonetti)

The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.

The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number. .

Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).

The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.

Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

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

An integer $n > 1$ has the following property: for every (positive) divisor $d$ of $n, d + 1$ is a divisor of $n + 1$. Prove that $n$ is prime.

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

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$