Found problems: 364
Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$
[size=75]$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.[/size]
Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers
$N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.
Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.
Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.
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 $2021$ .
Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence?
(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number.
(b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.
Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.
There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression.
Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.
Consider the following equation in $x$: $$ax (x^2 + ax + 1) = b (x^2 + b + 1).$$ It is known that $a, b$ are real such that $ab <0$ and furthermore the equation has exactly two integer roots positive. Prove that under these conditions $a^2 + b^2$ is not a prime number.
Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$
Let $p$ be a prime number and $a$ be an integer.
Prove that if $2^p +3^p = a^n$ for some integer $n$, then $n = 1$.
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then
$$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$
is not an integer.
Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$.
[Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.]
[hide=Hint]$n$ is squarefree[/hide]
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 the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.
Find all prime numbers with $n \ge 3$ digits, having the property:
for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.
Let \(n > 1\) be an integer and \(p\) be a prime. Prove that if \(n|p-1\) and \(p|n^3-1\), then \(4p-3\) is a perfect square.
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.