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A346721
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Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-7)), 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.
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5
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0, 2, 3, 5, 6, 11, 17, 19, 21, 23, 28, 30, 32, 45, 47, 57, 59, 61, 63, 65, 67, 69, 76, 78, 80, 82, 84, 103, 107, 121, 125, 127, 129, 131, 135, 137, 139, 148, 150, 152, 156, 158, 160, 187, 189, 211, 213
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OFFSET
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1,2
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COMMENTS
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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(-7) or z = (x+0.5) + (y+0.5)*sqrt(-7) 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. When the latter are adjusted to make them regular, it 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.
General properties of the related hexagonal spiral sequences: (Start)
R is one of 7 rings where hexagons are appropriate. Each has elements of the form x + y*sqrt(-p) and (x+0.5) + (y+0.5)*sqrt(-p), where p is a (rational) prime congruent to 3 modulo 4.
When mapping the grid cells to quadratic integers, it is often convenient to write the latter as a + w*b, where w = 0.5*(1+sqrt(-p)). Cell m on the spiral represents A307011(m) + w*A307012(m).
We can find the primes without advanced mathematics, using multiplication formulas and a sieve as explained below.
w^2 = w - c, where c = (p+1)/4 (which is an integer as p == 3 (mod 4)). So, in general, the product of a_1 + w*b_1 and a_2 + w*b_2 is (a_1*a_2 - c*b_1*b_2) + w*(a_1*b_2 + a_2*b_1 + b_1*b_2). The norm (absolute square) of a + w*b is a^2 + a*b + c*b^2.
For k >= 1, the algebraic integers represented by cells numbered 3k*(k-1)+1 to 3k*(k+1) on the spiral (cells A003215(k-1) to A028896(k)) are positioned along a hexagon in the complex plane; they include rational integers k and -k, and have norms in the range [k^2*(4c-1)/4c, k^2*c] = [k^2*p/(p+1), k^2*c].
To determine the primes we may list the ring elements in an order such that they have nondecreasing norm, and use a sieve to remove the products of nonunits. So, we are only interested in elements with norm greater than 1 (i.e. nonzero, nonunit). At each round of sieving we note the first element, z, whose products we have not yet removed, and remove in turn the product of z and each element from z onwards in the list.
(End)
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REFERENCES
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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.
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LINKS
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FORMULA
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m is a term if and only if A345764(m) is a term.
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EXAMPLE
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Table showing derivation of initial terms.
The ring element, z, represented by spiral cell m is shown in the form A307011(m) + A307012(m)*w, where w = 0.5*(1+sqrt(-7)).
The column headed "(x,y)" gives x and y when z is written in the form z = x + y*sqrt(-7).
m | | z (x,y) status n a(n)=m
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0 0 0 0 ( 0.0, 0.0) zero 1 0
1 1 0 1 ( 1.0, 0.0) unit
2 0 1 w ( 0.5, 0.5) prime 2 2
3 -1 1 -1+w (-0.5, 0.5) prime 3 3
4 -1 0 -1 (-1.0, 0.0) unit
5 0 -1 -w (-0.5,-0.5) prime 4 5
6 1 -1 1-w ( 0.5,-0.5) prime 5 6
7 2 -1 2-w ( 1.5,-0.5) = -w*w
8 2 0 2 ( 2.0, 0.0) = (1-w)*w
9 1 1 1+w ( 1.5, 0.5) = (1-w)*(w-1)
10 0 2 2w ( 1.0, 1.0) = 2*w
11 -1 2 -1+2w ( 0.0, 1.0) prime 6 11
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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