A345437 Represent the ring R = {x+y*sqrt(-2): x, y rational integers} by the cells centered at the points (x,y) of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells at (0,0) and (1,0) numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in R.

0, 2, 3, 4, 6, 7, 8, 25, 26, 28, 29, 32, 34, 37, 38, 40, 41, 44, 46, 57, 63, 73, 79

Offset

1,2

Comments

R is the ring of integers in the quadratic number field Q(sqrt(-2)). The element x+y*sqrt(-2) in R has norm x^2+2*y^2.

A033715 gives the number of elements in R with norm n.

There are two units, +-1, of norm 1.

A341784 gives the norms of the primes in R, and A345438 gives the numbers of primes of those norms.

References

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

Table of n, a(n) for n=1..23.

N. J. A. Sloane, Illustration of initial terms [An enlargement of Figure 3 of Wichmann (2019), showing the numbering of the initial cells of the square spiral. The origin is black, the two units +-1 are red, and the primes are blue.]

Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019

Brian Wichmann, Detail of Figure 3 from the previous link

Example

One can read off the primes from the blue cells in the illustration. The first few primes are +-sqrt(-2), 2 of norm 2; +-1+-sqrt(-2), 4 of norm 3; +-3+-sqrt(-2), 4 of norm 11; ... (see A345438).

Crossrefs

Cf. A003173, A033715, A341784, A345435, A345436, A345438.

Sequence in context: A091421 A138936 A135604 * A084826 A118944 A132988

Adjacent sequences: A345434 A345435 A345436 * A345438 A345439 A345440

Keyword

nonn

Author

N. J. A. Sloane, Jun 23 2021

Status

approved