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
Keywords
Examples
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).
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
- 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
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