A345436 Represent the ring of Gaussian integers E = {x+y*i: x, y rational integers, i = sqrt(-1)} by the cells of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells representing the ring identities 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

0, 2, 4, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 59, 61, 67, 69, 75, 77, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157

Offset: 1

Author

N. J. A. Sloane, Jun 23 2021

Keywords

• nonn

Comments

The cell with spiral index m represents the Gaussian integer A174344(m+1) + A274923(m+1) * i. So the set of Gaussian primes is {A174344(a(n)+1) + A274923(a(n)+1) * i : n >= 2}. - Peter Munn, Aug 02 2021

The Gaussian integer z = x+i*y has norm x^2+y^2. There are four units (of norm 1), +-1, +-i. The number of Gaussian integers of norm n is A004018(n).

The norms of the Gaussian primes are listed in A055025, and the number of primes with a given norm is given in A055026.

The successive norms of the Gaussian integers along the square spiral are listed in A336336.

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.2, p. 106.
• L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
• 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

• Eric Weisstein's World of Mathematics, Gaussian prime.
• Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Fig. 2.

Crossrefs

Equals A308412 - 1. Cf. A345435, A345437.

Cf. also A003173, A004018, A055025, A055026, A103431, A174344, A274923, A336336.

Extensions

Name clarified by Peter Munn, Aug 02 2021