Found problems: 916
Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$
Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that:
\[a^p+b^p=p^c.\]
Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.
Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.
Find all $(x,y,n) \in {\mathbb{N}}^3$ such that $\gcd(x, n+1)=1$ and $x^{n}+1=y^{n+1}$.
Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]
Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]
Find all primes $p, q$ satisfying the equation $2p^q - q^p = 7.$
Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.
Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.
Prove that the equation $ 2x^2 - 215y^2 = 1 $ has no integer solutions.
Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$.
Titu Andreescu, Mathematics Department, College of Texas, USA
There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.
Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.
Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$
Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$.
If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$.
[i]Proposed by Mark Sellke[/i]
Find all $a,b \in \mathbb{N}$ such that
$$2049^ba^{2048}-2048^ab^{2049}=1.$$
[i]Proposed by fattypiggy123 and 61plus[/i]
Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation
\[n^3+m^3-nm(n+m)=2023.\]
Prove:
(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$
(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$
Find all pairs $(x,y)$ of positive integers that satisfy
$$2^x+17=y^4$$.
Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$.
a) Prove $|a -3b|\ge 1$.
b) Find all pairs of positive integers $(a, b)$ satisfying the equality.