The page "Crible
géométrique (hyperbole)" by JeanPaul Davalaninspired
me to write the interactive Java applet below. On the rectangular hyperbola y = k/x (k>0, x natural number) mark two points P_{1}(x_{1} , y_{1}) and P_{2}(x_{2} , y_{2}), and draw the secant through P_{1} and P_{2}. In case of x_{1} = x_{2} = x draw the tangent of the hyperbola y =  k/x^{2} + 2k/x. Then from (k, 0) draw a second line perpendicular to the first, which will intersect the yaxis at (0, x_{1} · x_{2}). The points of intersection are (0, x_{1} · y_{1}) indicate the product, omitting the prime numbers {2, 3, 5, 7, 11, 13, 17, 19, 23,
29, 31, 37, 41, 43, 47, ...}
if x_{1}≠1
or x_{2}≠1.
The construction also interprets the multiplication of real numbers. 
Select the grid size (pixels). 

Select the the constant k of the
parabola y=k/x. 

Click two integers on the horizontal axis,
the cursor will change to cross hair. The first click is marked by a fat circle. 


Hold down the command key to add
subsequent clicks. 
If two points are marked the menu will
be enabled. 

Selecting "2 lines" will add the
points at x_{1}+1, x_{1}+2 selecting "3 lines" will add the points at x_{1}+1, x_{1}+2, x_{1}+3 and so on. 
The slope m of the first line
through P_{1} and P_{2}
is and the slope of the perpendicalar line
The equation of this line,
intersecting the vertical axis at y(0) = x_{1}
·
x_{1}
The first line (secant or
tangent) intersects the vertical axis at (0,
k[1/x_{1}
+ 1/x_{2}])
For the point of intersection of the two lines (x_{s}, y_{s}) we find: 

Crible géométrique (hyperbole) (JeanPaul Davalan) A Parabola Sieve for Prime Numbers (Wolfram) Catching primes (Abigail Kirk) 
20172023 J. Giesen
updated: 20.10.2023