Home: 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. 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 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 and the slope of the perpendicalar line The equation of this line, intersecting the vertical axis at  y(0) = x1 · x1 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: Web Links Crible géométrique (hyperbole) (Jean-Paul Davalan) A Parabola Sieve for Prime Numbers (Wolfram) Catching primes (Abigail Kirk)

2017  J. Giesen

updated: 2017, Feb 11

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