Found problems: 76
Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.
Let $K$ be a positive integer such that there exist a triple of positive integers $(x,y,z)$ such that
\[x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx\]
are all perfect cubes.
(a) Prove that $K \ne 2$ and $K \ne 4$
(b) Find the minimum value of $K$ that satisfies.
[i]Proposed by Muhammad Afifurrahman[/i]
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ .
Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners.
Prove that Germán can always achieve his goal.
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.
Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.
Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.
Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.
Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.
Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.
Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers