Found problems: 721
Prove that when dividing a prime number with $30$, remainder is always not a composite number
Let $n$ be a positive integer.
Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$.
[i]P. Erdos[/i]
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.
Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$
Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.
Let $N$ be the number of ordered pairs $(x,y)$ st $1 \leq x,y \leq p(p-1)$ and :
$$x^{y} \equiv y^{x} \equiv 1 \pmod{p}$$
where $p$ is a fixed prime number.
Show that :
$$(\phi {(p-1)}d(p-1))^2 \leq N \leq ((p-1)d(p-1))^2$$
where $d(n)$ is the number of divisors of $n$
Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.
Any rational number admits a non-decimal representation unlimited decimal expansion. This development has the particularity of being periodic.
Examples: $\frac{1}{7} = 0.142857142857…$ has a period $6$ while $\frac{1}{11}=0.0909090909 …$ $2$ periodic.
What are the reciprocals of the prime integers with a period less than or equal to five?
Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.
Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$
The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.
Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$
How many different prime numbers are factors of $ N$ if
\[ \log_2 (\log_3 (\log_5 (\log_7 N))) \equal{} 11?
\]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 7$
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?
(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3.
(b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.
Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.
Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?
Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.
For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?
Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.