A346725 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-67)) as explained in the comments. Number the cells along the counterclockwise hexagonal spiral that starts with cells 0 and 1 representing integers 0 and 1. List the cells that represent 0 or a prime in the ring.

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91, 95, 97, 101, 103, 107, 109

Offset: 1

Author

Peter Munn, Aug 23 2021

Keywords

• nonn

Comments

In this entry we use "rational integers" to refer to integers in their usual sense as whole numbers - they form a subset of the algebraic integers that form the ring, which we denote "R".

The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-67) or z = (x+0.5) + (y+0.5)*sqrt(-67) where x and y are rational integers. Plotted as points on a plane, they can be joined in a grid of isosceles triangles or be seen as the center points of hexagonal regions. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.

(To be precise, we map each element, z, to the region of the complex plane containing the points that have z as their nearest ring element, then map these (hexagonal) regions continuously to the cells of a (regular) hexagonal grid.)

R is one of 9 related rings that are unique factorization domains, meaning their elements factorize into prime elements in a unique way, just as with rational integers and prime numbers. See the Wikipedia link or the Stark reference, for example.

This set of sequences is inspired by tilings: see the Wichmann link. Each tiling represents one of the 9 rings and shows the primes as distinctively colored squares or hexagons as appropriate.

6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.

This sequence first differs from A346724 at a(70) = 103 <> A346724(70) = 109 and from A346726 at a(116) = 171 <> A346726(116) = 169.

The first rational prime not to appear as a cell number in this sequence is 191.

Examples

			The sequence is constructed in the same way as A346721, but the relevant prime is 67 instead of 7. See the example section of A346721.
		

References

• L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
• H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

Links

• OEIS Wiki, Algebraic integers.
• Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
• Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 9.
• Wikipedia, Quadratic integer.
• Wikipedia, Unique factorization domain.

Crossrefs

Cf. A003173, A345764.

Norms of primes in R: A341789.

Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346723 (D=-19), A346724 (D=-43), A346726 (D=-163).

Formula

m is a term if and only if A345764(m) is a term.