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Solve in non-negative integers the equation $$x^3 + 2y^3 = 4z^3$$

Solve in nonnegative integers the equation $$2^x -1 = xy$$


Solve in integers the equation $x^2+y^2+z^2-2xyz=0$

Show that the equation $x^4 + y^4 = z^2$ is not solvable in integers if $xyz\ne 0$.

Show that $\sqrt{2}$ is an irrational number.

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 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 $$x^3 + y^3 + z^3 = 3xyz$$