# Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:

[To see these pictures in higher resolution, you can try it out here and here. Note that these are two different zig-zag patterns (with different "lattice parameters").]

One sees diagonal bands with an above-average density of prime numbers. One of them contains the prime numbers $7, 11, 29, 43, 73, 89, 131, 157, 181, 211, 239, 379, 421, 461, 601,\dots$.

How can these bands be explained?

One also sees a rather distinct band from the lower left going upwards at an angle of roughly $75°$. It contains the prime numbers $229, 271, 317, 367, 421, 479, 541, 607, 677, 751, \dots$.

Note, that the vertical and horizontal borders of the pattern contain no primes at all, which can be easily understood.

[See my related question concerning the distributions of other numbers.]

By the way (and off the record): The detecting of lines in patterns (most easily of straight lines) is fun also in non-mathematical contexts (be them obvious or not):

• I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect. – Arthur Aug 17 '18 at 9:26
• Yes, in principal, more or less. But Ulam-like coverings of $\mathbb{Z}\times \mathbb{Z}$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $\mathbb{N}\times \mathbb{N}$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here). – Hans-Peter Stricker Aug 17 '18 at 17:11
• The first sequence you described corresponds to A081352 (polynomials $n^2+3n\pm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials. – Sil Aug 19 '18 at 17:14
• The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials) – Sil Aug 19 '18 at 17:45
• Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected? – Hans-Peter Stricker Aug 19 '18 at 17:55