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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?
\[ \large \color{blue}{x}^2-\color{red}{y}^2=2011\]
How many integral solutions \((\color{blue}{x},\color{red}{y})\) are there for the equation above?
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by
Vinay Sipani
\[ \begin{eqnarray} ab+cd&=&38 \\ ac+bd&=&34\\ ad+bc&=&43\\ \end{eqnarray} \]
Let \(a,b,c\) and \(d\) positive integers such that they satisfy the system of equations above. Find \(a+b+c+d\).
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by
Paola Ramírez
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.
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by
Calvin Lin
\[\large \frac{1}{x}+\frac{1}{y}=\frac{1}{156}\]
Find the number of ordered pairs of positive integers \((x, y) \) that satisfy the equation above.
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by
Vaibhav Prasad
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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by
William Lockhart