Show that $x^4 + y^4 = z^2$ is not solvable in positive integers.
Find all primes $p$ for which there exist positive integers $x$, $y$, and $n$ such that $$p^n = x^3+y^3$$
Let $a_1, a_2, \cdots, a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \cdots = a_{2n+1}$.
Prove that if positive integer $a$ and $b$ are such that $ab+1$ divides $a^2 + b^2$. then $$\frac{a^2+b^2}{ab+1}$$ is a square number.
Solve in positive integers $x^2 + y^2 + x+y+1 = xyz$
Solve in positive integers $x$, $y$, $u$, $v$ the system of equations
$$
\left\{
\begin{array}{ll}
x^2 +1 &= uy\\
y^2 + 1&= vx
\end{array}
\right.
$$
Show that if there is a triple $(x, y, z)$ of positive integers such that $$x^2 +y^2 +1 = xyz$$
then $z=3$, and find all such triples.
Find all solutions of $a^3 + b^3 = 2(s^2+t^2)$
Solve in integers $x^2 + y^2 +z^2 - 2xyz=0$.
Show that the sides of a Pythagorean triangle in which the hypotenuse exceeds the larger leg by 1 are given by $\frac{n^2-1}{2}$, $n$ and $\frac{n^2+1}{2}$
Show that if the lengths of all the three sides in a right triangle are whole numbers, then radius of its incircle is always a whole number too.
Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$
Find all the Pythagorean triangles whose two sides are consecutive integers.
Find all the positive integer triplets $(m, n, k)$ that satisfy the equation $$1!+2!+3!+\cdots+m!=n^k$$
where $m, n , k > 1$
Solve in positive integers the equation $x^2 + y^2 = z^4$, where $\gcd(x,y)=1$ and $x$ is even.
Show that the sum and difference of two squares cannot be both squares themselves.
If for a given positive integer $n$, the equation $x^n + y^n = z^n$ is not solvable in positive integer. Show that the equation $$x^{2n} + y^{2n} = z^{2n}$$
is not solvable in positive integers either.
Show that the equation $$x^2 + y^2 -19xy - 19 =0$$ is not solvable in integers.
Solve in positive integers $$x^3 + y^3 + z^3 = 3xyz$$
Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$
Solve in integers the equation $$(x+y)^2 = x^3 + y^3$$
Solve the following equation in positive integers: $15x - 35y + 3 = z^2$
Solve the following equation in positive integers: $3\times (5x + 1)=y^2$
Find all pairs of integers $(x, y)$ such that $5\times (x^2 + 3)= y^2$.
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1$, $k_2$, $\dots$ , $k_n$ for which $$k_1^2 + k_2^2 + \cdots +k_n^2 = 2002$$