Number Theory

# Quadratic Diophantine Equations: Level 3 Challenges

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

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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