## Playing with the Matiyasevich-Stechkin visual sieve

Back in 1999, Matiyasevich and Stechkin proposed to visualize the sieving process that singles out the primes by using a parabola, which is fairly natural in hindsight. Here is a graph from a geogebra version :

That puts the sieve in the realm of Euclidean geometry, and it is very tempting to play with this construction a little bit. Here are some half-baked ideas, feel free to mention better ones…

One could try to map the primes so obtained by various transformations associated naturally to a parabola :

• trying inversions (in particular through the vertex, and through the focus) for various values of the inversion radius would result in the primes being compactified to various line segments, so not something terribly insightful nor of artistic merit
• the same would occur if looking at the inverse of the projection from a circle
• in a different direction, one could of course move the parabola around within conic sections, resulting in lots of elliptic (compact) and hyperbolic (non-compact) images of the set of primes [this has very probably been considered before, any reference and keyword would be welcome!]

Another line of thought would be to set up some sort of billiard dynamics (or indeed wave dynamics) inside the parabola, e.g. sending light rays of pulses from each prime and looking at the resulting interference patterns over time, that might lead to some artistically pleasing visual patterns (an idea that might be explored at a later date)

A final thing that comes to mind is to try and construct constants that encapsulate some special features from the complete construction. One possibility I’ve looked at is to add the areas of the quadrilaterals that surround each prime, but if I got this right the sum is $2\sqrt{2} \sum_{p\in\mathbb{P},\ p\geq 7}\frac{p-1}{(p-2)(p-3)}$, which unfortunately diverges…