Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$

Find all pairs of prime numbers $p\,,q$ for which: \[p^2 \mid q^3 + 1 \,\,\, \text{and} \,\,\, q^2 \mid p^6-1\] [i]Proposed by P. Boyvalenkov[/i]

Prove that no term of the sequence $10001$, $100010001$, $1000100010001$ , $...$ is prime.

Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.

Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.

Find all prime numbers $p$ for which $x^4\equiv -1\, (mod\, p)$ has a solution.

Let $p$ be a prime. Prove that $p^2+p+1$ is never a perfect cube.

Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.

There exists a block of $1000$ consecutive positive integers containing no prime numbers, namely, $1001!+2$, $1001!+3$, $\cdots$, $1001!+1001$. Does there exist a block of $1000$ consecutive positive integers containing exactly five prime numbers?

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

There are finitely many primes dividing the numbers $\{ a \cdot b^n + c\cdot d^n : n=1, 2, 3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.

Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

The sequence $\{a_n\}$ is defined by $a_0 = 1, a_1 = 2,$ and for $n \geq 2,$ $$a_n = a_{n-1}^2 + (a_0a_1 \dots a_{n-2})^2.$$ Let $k$ be a positive integer, and let $p$ be a prime factor of $a_k.$ Show that $p > 4(k-1).$

For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$? [i]Russian Competition[/i]

Let $p_1<p_2<p_3<p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers, such that $p_4 - p_1 = 8$ and $q_4 - q_1= 8$. Suppose $p_1 > 5$ and $q_1>5$. Prove that $30$ divides $p_1 - q_1$.

Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$ where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$. Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23. [i]Author: K. Kokhas[/i]

In the diagram below, fill the $12$ circles with numbers from the following bank so that each number is used once. Two circles connected by a single line must contain relatively prime numbers. Two circles connected by a double line must contain numbers that are not relatively prime. $$\text{Bank: } 20, 21, 22, 23, 24, 25, 27, 28, 30 ,32, 33 ,35$$ [asy] real HRT3 = sqrt(3) / 2; void drawCircle(real x, real y, real r) { path p = circle((x,y), r); draw(p); fill(p, white); } void drawCell(int gx, int gy) { real x = 0.5 * gx; real y = HRT3 * gy; drawCircle(x, y, 0.35); } void drawEdge(int gx1, int gy1, int gx2, int gy2, bool doubled) { real x1 = 0.5 * gx1; real y1 = HRT3 * gy1; real x2 = 0.5 * gx2; real y2 = HRT3 * gy2; if (doubled) { real dx = x2 - x1; real dy = y2 - y1; real ox = -0.035 * dy / sqrt(dx * dx + dy * dy); real oy = 0.035 * dx / sqrt(dx * dx + dy * dy); draw((x1+ox,y1+oy)--(x2+ox,y2+oy)); draw((x1-ox,y1-oy)--(x2-ox,y2-oy)); } else { draw((x1,y1)--(x2,y2)); } } drawEdge(2, 0, 4, 0, true); drawEdge(2, 0, 1, 1, true); drawEdge(2, 0, 3, 1, true); drawEdge(4, 0, 3, 1, false); drawEdge(4, 0, 5, 1, false); drawEdge(1, 1, 0, 2, false); drawEdge(1, 1, 2, 2, false); drawEdge(1, 1, 3, 1, false); drawEdge(3, 1, 2, 2, true); drawEdge(3, 1, 4, 2, true); drawEdge(3, 1, 5, 1, false); drawEdge(5, 1, 4, 2, true); drawEdge(5, 1, 6, 2, false); drawEdge(0, 2, 1, 3, false); drawEdge(0, 2, 2, 2, false); drawEdge(2, 2, 1, 3, false); drawEdge(2, 2, 3, 3, true); drawEdge(2, 2, 4, 2, false); drawEdge(4, 2, 3, 3, false); drawEdge(4, 2, 5, 3, false); drawEdge(4, 2, 6, 2, false); drawEdge(6, 2, 5, 3, true); drawEdge(1, 3, 3, 3, true); drawEdge(3, 3, 5, 3, false); drawCell(2, 0); drawCell(4, 0); drawCell(1, 1); drawCell(3, 1); drawCell(5, 1); drawCell(0, 2); drawCell(2, 2); drawCell(4, 2); drawCell(6, 2); drawCell(1, 3); drawCell(3, 3); drawCell(5, 3); [/asy]