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 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 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 .
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 , for example, could also be written as . 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 . 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 . This here is a composition of sequences and .
Factorizations
+ Norm 1–250 Norm Integer Factorization 2 (p) 4 5 (p) (p) 8 9 (p) 10 13 (p) (p) 16 17 (p) (p) 18 20 25 26 29 (p) (p) 32 34 36 37 (p) (p) 40 41 (p) (p) 45 49 (p) 50 52 53 (p) (p) 58 61 (p) (p) 64 65 68 72 73 (p) (p) 74 80 81 82 85 89 (p) (p) 90 97 (p) (p) 98 100 101 (p) (p) 104 106 109 (p) (p) 113 (p) (p) 116 117 121 (p) 122 125 128 130 136 137 (p) (p) 144 145 146 148 149 (p) (p) 153 157 (p) (p) 160 162 164 169 170 173 (p) (p) 178 180 181 (p) (p) 185 193 (p) (p) 194 196 197 (p) (p) 200 202 205 208 212 218 221 225 226 229 (p) (p) 232 233 (p) (p) 234 241 (p) (p) 242 244 245 250 + Norm 251–500 Norm Integer Factorization 256 257 (p) (p) 260 261 265 269 (p) (p) 272 274 277 (p) (p) 281 (p) (p) 288 289 290 292 293 (p) (p) 296 298 305 306 313 (p) (p) 314 317 (p) (p) 320 324 325 328 333 337 (p) (p) 338 340 346 349 (p) (p) 353 (p) (p) 356 360 361 (p) 362 365 369 370 373 (p) (p) 377 386 388 389 (p) (p) 392 394 397 (p) (p) 400 401 (p) (p) 404 405 409 (p) (p) 410 416 421 (p) (p) 424 425 433 (p) (p) 436 441 442 445 449 (p) (p) 450 452 457 (p) (p) 458 461 (p) (p) 464 466 468 477 481 482 484 485 488 490 493 500 + Norm 501–750 Norm Integer Factorization 505 509 (p) (p) 512 514 520 521 (p) (p) 522 529 (p) 530 533 538 541 (p) (p) 544 545 548 549 554 557 (p) (p) 562 565 569 (p) (p) 576 577 (p) (p) 578 580 584 585 586 592 593 (p) (p) 596 601 (p) (p) 605 610 612 613 (p) (p) 617 (p) (p) 625 626 628 629 634 637 640 641 (p) (p) 648 650 653 (p) (p) 656 657 661 (p) (p) 666 673 (p) (p) 674 676 677 (p) (p) 680 685 689 692 697 698 701 (p) (p) 706 709 (p) (p) 712 720 722 724 725 729 730 733 (p) (p) 738 740 745 746 + Norm 751–1000 Norm Integer Factorization 754 757 (p) (p) 761 (p) (p) 765 769 (p) (p) 772 773 (p) (p) 776 778 784 785 788 793 794 797 (p) (p) 800 801 802 808 809 (p) (p) 810 818 820 821 (p) (p) 829 (p) (p) 832 833 841 842 845 848 850 853 (p) (p) 857 (p) (p) 865 866 872 873 877 (p) (p) 881 (p) (p) 882 884 890 898 900 901 904 905 909 914 916 922 925 928 929 (p) (p) 932 936 937 (p) (p) 941 (p) (p) 949 953 (p) (p) 954 961 (p) 962 964 965 968 970 976 977 (p) (p) 980 981 985 986 997 (p) (p) 1000
See also Gaussian integer Table of divisors Integer factorization
References
External links
OEIS: Gaussian Primes