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Quadratic Diophantine Equations

The Pythagorean Theorem is one of the most famous quadratic diophantine equations, which are equations with squared variables solved over the integers. How many solutions can you find?

Level 3

\[ \large \color{blue}{x}^2-\color{red}{y}^2=2011\]

How many integral solutions \((\color{blue}{x},\color{red}{y})\) are there for the equation above?

\[ \begin{eqnarray} ab+cd&=&38 \\ ac+bd&=&34\\ ad+bc&=&43\\ \end{eqnarray} \]

Let \(a,b,c\) and \(d\) positive integers such that they satisfy the system of equations above. Find \(a+b+c+d\).

Find the number of ordered quadruples of positive integers \((x,y,p,q)\) satisfying

\[x^3y-xy^3=pq,\]

and \(p,q\) are prime numbers.

This problem is posed by Daniel C.

\[\large \frac{1}{x}+\frac{1}{y}=\frac{1}{156}\]

Find the number of ordered pairs of positive integers \((x, y) \) that satisfy the equation above.

How many ordered tuples of natural numbers \((a,b,c)\) satisfy \[\frac{1}{a}+\frac{1}{ab}+\frac{1}{abc}=1?\]

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