View Single Post
Mar23-11, 01:57 AM
P: 688
Right; your parabolas do not pass through the origin, instead they have been shifted so that the parabola representing the multiples of n passes through the point in the first parabola that represents the integer n. (This way, the horizontal lines will only intersect true multiples of n, clearing up other instances of n itself.)

A similar thing can be done by shifting the lines I mentioned before; the line with slope n would pass not through the origin, but through the point (n,n) on the first line. Attached is a drawing.

In fact, graphs of any monotonic curve (x^2, x^3, exp x, ln x, ...) would also produce the primes in the same manner (namely, in the manner of Erathostenes' sieve).

Edit: My bad, x^2 is not, overall, monotonic. I was referring to curves that are increasingly monotonic on the first quadrant; that is, for x>0, whenever y>x you have f(y)>f(x), so that the vertical ordering of the points is preserved.
Attached Thumbnails