Table of Gaussian integer factorizations
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes.
Note that there are rational primes which are not Gaussian primes. A simple example is the rational prime 5, which is factored as 5=(2+i)(2−i) in the table, and therefore not a Gaussian prime.
Conventions
The second column of the table contains only integers in the first quadrant, which means the real part x is positive and the imaginary part y is non-negative. The table might have been further reduced to the integers in the first octant of the complex plane using the symmetry y + ix =i (x − iy).
The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. The entry 4+2i = −i(1+i)2(2+i), for example, could also be written as 4+2i= (1+i)2(1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane with absolute value of the real part larger than or equal to the absolute value of the imaginary part.
The entries are sorted according to increasing norm x2 + y2 (sequence A001481 in the OEIS). The table is complete up to the maximum norm at the end of the table in the sense that each composite or prime in the first quadrant appears in the second column.
Gaussian primes occur only for a subset of norms, detailed in sequence OEIS: A055025. This here is a composition of sequences OEIS: A103431 and OEIS: A103432.
Factorizations
See also
References
- Dresden, Greg; Dymacek, Wayne (2005). "Finding factors of factor rings over the Gaussian integers". American Mathematical Monthly. 112 (7): 602–611. doi:10.2307/30037545. JSTOR 30037545. MR 2158894.
- Gethner, Ellen; Wagner, Stan; Wick, Brian (1998). "A stroll through the Gaussian primes". Amer. Math. Monthly. 105 (4): 327–337. doi:10.2307/2589708. JSTOR 2589708. MR 1614871.
- Matsui, Hajime (2000). "A bound for the least Gaussian prime omega with alpha < arg(omega) < beta". Arch. Math. 74 (6): 423–431. doi:10.1007/s000130050463. MR 1753540.