Found problems: 23
Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.
From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .
Find all values of the positive integer $k$ that has the property:
There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.
Prove that number $2^{2008}\cdot2^{2010}+5^{2012}$ is not prime
Let $m$ be a positive integer. Show that there exists a positive integer $n$ such that each of the $2m+1$ integers
$$ 2^{n}-m,2^{n}-(m-1),\ldots,2^{n}+(m-1),2^{n}+m$$
is positive and composite.
Show that there are infinitely many positive integers $a$ such that the number $n^4+a$ is composite for every positive integer $n.$ Give 5 (different) numbers $a$ with the mentioned property.
$2000$ numbers are considered: $11, 101, 1001, . . $. Prove that at least $99\%$ of these numbers are composite.
Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.
For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?
Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite.
(V Senderov)
Define a sequence $(a_n)$ by $a_0 =-4 , a_1 =-7$ and $a_{n+2}= 5a_{n+1} -6a_n$ for $n\geq 0.$ Prove that there are infinitely many positive integers $n$ such that $a_n$ is composite.
Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?
Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ .
Prove that for any integer $a > 1$ there is a prime $p$ for which $1+a+a^2+...+ a^{p-1}$ is composite.
Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions:
i) the number $n$ is composite;
ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.
The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.
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.
Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$.
Show that at least one between $a - 1, b - 1, c -1$ is composite number.
Let $a, b, c, d$ be positive integers such that $ab = cd$. Prove that $w = a^{2006} + b^{2006} + c^{2006} + d^{2006}$ is composite.
Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$ there exist at least $k$ consecutive composite integers in the set
{$n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$}.
Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$
We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime
Show that there are no integers $n$ such that $n^4 + 4^n$ is a prime greater than $5$.