This is a variation on a theme by Ulam. It is well-known that Ulam’s spiral is related to the high density of primes along some quadratic polynomials, as per Hardy & Littlewood’s conjecture F.

What about other curves, in particular space-filling curves like that of Peano or that of Hilbert ? Would one expect to see particular patterns ?

Well, not knowing what to expect, I’ve tried to look at Hilbert’s. Below are thus primes marked on iterations 7 and 8 of Hilbert’s curve, with also primes shown without the curve. (I’ve chosen the convention that has the first vertex at the top left corner, and I prefer not to show the quick and ugly code.)

End result : unfortunately I can’t quite spot anything too noticeable (that the primes avoid some diagonals is an easy consequence of the curve being built from units of 4 vertices, but beyond that…). Also, for some reason worpress.com wouldn’t allow my svg files, so these are uglier png versions…

**Edit (3rd august 2015):** since primes do rarefy, it was tempting to look at a few more iterations and see if at least there are more around the top left corner on a larger scale. This is indeed seen in the plots of iteration 10 and 11 that I’ve added after the original ones. Iteration 11 contains 4,194,304 integers, and thus includes a fair amount of primes. But apart from the top left corner denser region the plots in fact look really uniform, despite lots of not too small prime gaps in these ranges, so there’s really no pattern there.

And now iteration 10 and 11 (making the points on the 10th somewhat larger for better viewing).