Number Theory

Quadratic Diophantine Equations

Quadratic Diophantine Equations: Level 3 Challenges

         

x2y2=2011 \large \color{#3D99F6}{x}^2-\color{#D61F06}{y}^2=2011

How many integral solutions (x,y)(\color{#3D99F6}{x},\color{#D61F06}{y}) are there for the equation above?

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

Let a,b,ca,b,c and dd positive integers such that they satisfy the system of equations above. Find a+b+c+da+b+c+d.

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

x3yxy3=pq,x^3y-xy^3=pq,

and p,qp,q are prime numbers.

This problem is posed by Daniel C.

1x+1y=1156\large \frac{1}{x}+\frac{1}{y}=\frac{1}{156}

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

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

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