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Sieve for Prime Numbers by a Rectangular Hyperbola
The page "Crible géométrique (hyperbole)" by Jean-Paul Davalaninspired me to write the interactive Java applet below.

On the rectangular hyperbola y = k/x  (k>0,  x natural number) mark two points P1(x1 , y1) and
P2(x2 , y2), and draw the secant through P1 and P2. In case of x1 = x2 = x  draw the tangent of the hyperbola
y = - k/x2 + 2k/x.
Then from (-k, 0) draw a second line perpendicular to the first, which will intersect the y-axis
at (0, x1 · x2).

The points of intersection are (0, x1 · y1) indicate the product, omitting the prime numbers
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...}
if x1≠1 or x2≠1.

The construction also interprets the multiplication of real numbers.


grid
 Select the grid size (pixels).
constant
 Select the the constant k of the parabola  y=k/x.
first number
 Click two integers on the horizontal axis, the cursor will change to cross hair.
The first click is marked by a fat circle.
keys
subsequent
 Hold down the command key to add subsequent clicks.
line menu
 If two points are marked the menu will be enabled.
lines
 Selecting "2 lines" will add the points  at x1+1, x1+2

selecting "3 lines" will add the points at x1+1, x1+2,  x1+3

and so on.


The slope m of the first line through P1 and P2 is
hyperbola prime numbers

and the slope of the perpendicalar line
hyperbola sieve for prime numbers

The equation of this line, intersecting the vertical axis at  y(0) = x1 · x1
a sieve for prime numbers

The first line (secant or tangent) intersects the vertical axis at (0, k[1/x1 + 1/x2])

For the point of intersection of the two lines (xs, ys) we find:

intersection




 
Web Links

Crible géométrique (hyperbole) (Jean-Paul Davalan)

A Parabola Sieve for Prime Numbers (Wolfram)

Catching primes (Abigail Kirk)


2017-2023  J. Giesen

updated: 20.10.2023

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