Found problems: 14
Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
Show that there are infinitely many positive integers $a$ that cannot be written as $a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},$ where the $a_i$ are positive integers. State and prove a generalization.
Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.
Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers
An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.
a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$
b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.
(I. Voronovich)
Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?
The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$.
(a) Show that there are infinitely many good numbers.
(b) Show that if $n$ is a good number, then $7 \nmid n$.
Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$
Consider the equation $x^3 + y^3 + z^3 = 2$.
a) Prove that it has infinitely many integer solutions $x,y,z$.
b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.
Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.
Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$
a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares.
b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.
We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime