Found problems: 364
Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.
Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is:
(A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.
If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is
(A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.
Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
Find all natural numbers $n$ for which both $n^n + 1$ and $(2n)^{2n} + 1$ are prime numbers.
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.
For primes $a, b,c$ that satis fy the following, calculate $abc$.
$\bullet$ $b + 8$ is a multiple of $a$,
$\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$
$\bullet$ $b + c = a^2 - 1$.
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$
Are there infinitely many different natural numbers $a_1,a_2, a_3,...$ so that for every integer $k$ only finitely many of the numbers $a_1 + k$,$a_2 + k$,$a_3 + k$,$...$ are numbers prime?
In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.
The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime.
Show that at least one of these numbers is prime.
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.
There are 168 primes below 1000. Then sum of all primes below 1000 is
[list=1]
[*] 11555
[*] 76127
[*] 57298
[*] 81722
[/list]
For two different prime numbers $p, q$, defi ne $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.
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.
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have?
[i](Zhuge Liang)[/i]
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} $
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]