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 human-readable[clarification needed] version of sequences OEIS: A103431 and OEIS: A103432.
Factorizations
| norm | integer | factors |
|---|---|---|
| 2 | 1+i | (p) |
| 4 | 2 | −i·(1+i)2 |
| 5 | 2+i 1+2i |
(p) (p) |
| 8 | 2+2i | −i·(1+i)3 |
| 9 | 3 | (p) |
| 10 | 1+3i 3+i |
(1+i)·(2+i) (1+i)·(2−i) |
| 13 | 3+2i 2+3i |
(p) (p) |
| 16 | 4 | −(1+i)4 |
| 17 | 1+4i 4+i |
(p) (p) |
| 18 | 3+3i | (1+i)·3 |
| 20 | 2+4i 4+2i |
(1+i)2·(2−i) −i·(1+i)2·(2+i) |
| 25 | 3+4i 4+3i 5 |
(2+i)2 i·(2−i)2 (2+i)·(2−i) |
| 26 | 1+5i 5+i |
(1+i)·(3+2i) (1+i)·(3−2i) |
| 29 | 2+5i 5+2i |
(p) (p) |
| 32 | 4+4i | −(1+i)5 |
| 34 | 3+5i 5+3i |
(1+i)·(4+i) (1+i)·(4−i) |
| 36 | 6 | −i·(1+i)2·3 |
| 37 | 1+6i 6+i |
(p) (p) |
| 40 | 2+6i 6+2i |
−i·(1+i)3·(2+i) −i·(1+i)3·(2−i) |
| 41 | 4+5i 5+4i |
(p) (p) |
| 45 | 3+6i 6+3i |
i·(2−i)·3 (2+i)·3 |
| 49 | 7 | (p) |
| 50 | 1+7i 5+5i 7+i |
i·(1+i)·(2−i)2 (1+i)·(2+i)·(2−i) −i·(1+i)·(2+i)2 |
| 52 | 4+6i 6+4i |
(1+i)2·(3−2i) −i·(1+i)2·(3+2i) |
| 53 | 2+7i 7+2i |
(p) (p) |
| 58 | 3+7i 7+3i |
(1+i)·(5+2i) (1+i)·(5−2i) |
| 61 | 5+6i 6+5i |
(p) (p) |
| 64 | 8 | i·(1+i)6 |
| 65 | 1+8i 4+7i 7+4i 8+i |
i·(2+i)·(3−2i) (2+i)·(3+2i) i·(2−i)·(3−2i) (2−i)·(3+2i) |
| 68 | 2+8i 8+2i |
(1+i)2·(4−i) −i·(1+i)2·(4+i) |
| 72 | 6+6i | −i·(1+i)3·3 |
| 73 | 3+8i 8+3i |
(p) (p) |
| 74 | 5+7i 7+5i |
(1+i)·(6+i) (1+i)·(6−i) |
| 80 | 4+8i 8+4i |
−i·(1+i)4·(2−i) −(1+i)4·(2+i) |
| 81 | 9 | 32 |
| 82 | 1+9i 9+i |
(1+i)·(5+4i) (1+i)·(5−4i) |
| 85 | 2+9i 6+7i 7+6i 9+2i |
i·(2−i)·(4+i) i·(2−i)·(4−i) (2+i)·(4+i) (2+i)·(4−i) |
| 89 | 5+8i 8+5i |
(p) (p) |
| 90 | 3+9i 9+3i |
(1+i)·(2+i)·3 (1+i)·(2−i)·3 |
| 97 | 4+9i 9+4i |
(p) (p) |
| 98 | 7+7i | (1+i)·7 |
| 100 | 6+8i 8+6i 10 |
−i·(1+i)2·(2+i)2 (1+i)2·(2−i)2 −i·(1+i)2·(2+i)·(2−i) |
| 101 | 1+10i 10+i |
(p) (p) |
| 104 | 2+10i 10+2i |
−i·(1+i)3·(3+2i) −i·(1+i)3·(3−2i) |
| 106 | 5+9i 9+5i |
(1+i)·(7+2i) (1+i)·(7−2i) |
| 109 | 3+10i 10+3i |
(p) (p) |
| 113 | 7+8i 8+7i |
(p) (p) |
| 116 | 4+10i 10+4i |
(1+i)2·(5−2i) −i·(1+i)2·(5+2i) |
| 117 | 6+9i 9+6i |
i·3·(3−2i) 3·(3+2i) |
| 121 | 11 | (p) |
| 122 | 1+11i 11+i |
(1+i)·(6+5i) (1+i)·(6−5i) |
| 125 | 2+11i 5+10i 10+5i 11+2i |
(2+i)3 i·(2+i)·(2−i)2 (2+i)2·(2−i) i·(2−i)3 |
| 128 | 8+8i | i·(1+i)7 |
| 130 | 3+11i 7+9i 9+7i 11+3i |
i·(1+i)·(2−i)·(3−2i) (1+i)·(2−i)·(3+2i) (1+i)·(2+i)·(3−2i) −i·(1+i)·(2+i)·(3+2i) |
| 136 | 6+10i 10+6i |
−i·(1+i)3·(4+i) −i·(1+i)3·(4−i) |
| 137 | 4+11i 11+4i |
(p) (p) |
| 144 | 12 | −(1+i)4·3 |
| 145 | 1+12i 8+9i 9+8i 12+i |
i·(2−i)·(5+2i) (2+i)·(5+2i) i·(2−i)·(5−2i) (2+i)·(5−2i) |
| 146 | 5+11i 11+5i |
(1+i)·(8+3i) (1+i)·(8−3i) |
| 148 | 2+12i 12+2i |
(1+i)2·(6−i) −i·(1+i)2·(6+i) |
| 149 | 7+10i 10+7i |
(p) (p) |
| 153 | 3+12i 12+3i |
i·3·(4−i) 3·(4+i) |
| 157 | 6+11i 11+6i |
(p) (p) |
| 160 | 4+12i 12+4i |
−(1+i)5·(2+i) −(1+i)5·(2−i) |
| 162 | 9+9i | (1+i)·32 |
| 164 | 8+10i 10+8i |
(1+i)2·(5−4i) −i·(1+i)2·(5+4i) |
| 169 | 5+12i 12+5i 13 |
(3+2i)2 i·(3−2i)2 (3+2i)·(3−2i) |
| 170 | 1+13i 7+11i 11+7i 13+i |
(1+i)·(2+i)·(4+i) (1+i)·(2+i)·(4−i) (1+i)·(2−i)·(4+i) (1+i)·(2−i)·(4−i) |
| 173 | 2+13i 13+2i |
(p) (p) |
| 178 | 3+13i 13+3i |
(1+i)·(8+5i) (1+i)·(8−5i) |
| 180 | 6+12i 12+6i |
(1+i)2·(2−i)·3 −i·(1+i)2·(2+i)·3 |
| 181 | 9+10i 10+9i |
(p) (p) |
| 185 | 4+13i 8+11i 11+8i 13+4i |
i·(2−i)·(6+i) i·(2−i)·(6−i) (2+i)·(6+i) (2+i)·(6−i) |
| 193 | 7+12i 12+7i |
(p) (p) |
| 194 | 5+13i 13+5i |
(1+i)·(9+4i) (1+i)·(9−4i) |
| 196 | 14 | −i·(1+i)2·7 |
| 197 | 1+14i 14+i |
(p) (p) |
| 200 | 2+14i 10+10i 14+2i |
(1+i)3·(2−i)2 −i·(1+i)3·(2+i)·(2−i) −(1+i)3·(2+i)2 |
| 202 | 9+11i 11+9i |
(1+i)·(10+i) (1+i)·(10−i) |
| 205 | 3+14i 6+13i 13+6i 14+3i |
i·(2+i)·(5−4i) (2+i)·(5+4i) i·(2−i)·(5−4i) (2−i)·(5+4i) |
| 208 | 8+12i 12+8i |
−i·(1+i)4·(3−2i) −(1+i)4·(3+2i) |
| 212 | 4+14i 14+4i |
(1+i)2·(7−2i) −i·(1+i)2·(7+2i) |
| 218 | 7+13i 13+7i |
(1+i)·(10+3i) (1+i)·(10−3i) |
| 221 | 5+14i 10+11i 11+10i 14+5i |
i·(3−2i)·(4+i) (3+2i)·(4+i) i·(3−2i)·(4−i) (3+2i)·(4−i) |
| 225 | 9+12i 12+9i 15 |
(2+i)2·3 i·(2−i)2·3 (2+i)·(2−i)·3 |
| 226 | 1+15i 15+i |
(1+i)·(8+7i) (1+i)·(8−7i) |
| 229 | 2+15i 15+2i |
(p) (p) |
| 232 | 6+14i 14+6i |
−i·(1+i)3·(5+2i) −i·(1+i)3·(5−2i) |
| 233 | 8+13i 13+8i |
(p) (p) |
| 234 | 3+15i 15+3i |
(1+i)·3·(3+2i) (1+i)·3·(3−2i) |
| 241 | 4+15i 15+4i |
(p) (p) |
| 242 | 11+11i | (1+i)·11 |
| 244 | 10+12i 12+10i |
(1+i)2·(6−5i) −i·(1+i)2·(6+5i) |
| 245 | 7+14i 14+7i |
i·(2−i)·7 (2+i)·7 |
| 250 | 5+15i 9+13i 13+9i 15+5i |
(1+i)·(2+i)2·(2−i) i·(1+i)·(2−i)3 −i·(1+i)·(2+i)3 (1+i)·(2+i)·(2−i)2 |
| norm | integer | factors |
|---|---|---|
| 256 | 16 | (1+i)8 |
| 257 | 1+16i 16+i |
(p) (p) |
| 260 | 2+16i 8+14i 14+8i 16+2i |
(1+i)2·(2+i)·(3−2i) −i·(1+i)2·(2+i)·(3+2i) (1+i)2·(2−i)·(3−2i) −i·(1+i)2·(2−i)·(3+2i) |
| 261 | 6+15i 15+6i |
i·3·(5−2i) 3·(5+2i) |
| 265 | 3+16i 11+12i 12+11i 16+3i |
i·(2−i)·(7+2i) i·(2−i)·(7−2i) (2+i)·(7+2i) (2+i)·(7−2i) |
| 269 | 10+13i 13+10i |
(p) (p) |
| 272 | 4+16i 16+4i |
−i·(1+i)4·(4−i) −(1+i)4·(4+i) |
| 274 | 7+15i 15+7i |
(1+i)·(11+4i) (1+i)·(11−4i) |
| 277 | 9+14i 14+9i |
(p) (p) |
| 281 | 5+16i 16+5i |
(p) (p) |
| 288 | 12+12i | −(1+i)5·3 |
| 289 | 8+15i 15+8i 17 |
i·(4−i)2 (4+i)2 (4+i)·(4−i) |
| 290 | 1+17i 11+13i 13+11i 17+i |
i·(1+i)·(2−i)·(5−2i) (1+i)·(2+i)·(5−2i) (1+i)·(2−i)·(5+2i) −i·(1+i)·(2+i)·(5+2i) |
| 292 | 6+16i 16+6i |
(1+i)2·(8−3i) −i·(1+i)2·(8+3i) |
| 293 | 2+17i 17+2i |
(p) (p) |
| 296 | 10+14i 14+10i |
−i·(1+i)3·(6+i) −i·(1+i)3·(6−i) |
| 298 | 3+17i 17+3i |
(1+i)·(10+7i) (1+i)·(10−7i) |
| 305 | 4+17i 7+16i 16+7i 17+4i |
i·(2+i)·(6−5i) (2+i)·(6+5i) i·(2−i)·(6−5i) (2−i)·(6+5i) |
| 306 | 9+15i 15+9i |
(1+i)·3·(4+i) (1+i)·3·(4−i) |
| 313 | 12+13i 13+12i |
(p) (p) |
| 314 | 5+17i 17+5i |
(1+i)·(11+6i) (1+i)·(11−6i) |
| 317 | 11+14i 14+11i |
(p) (p) |
| 320 | 8+16i 16+8i |
−(1+i)6·(2−i) i·(1+i)6·(2+i) |
| 324 | 18 | −i·(1+i)2·32 |
| 325 | 1+18i 6+17i 10+15i 15+10i 17+6i 18+i |
(2+i)2·(3+2i) i·(2−i)2·(3+2i) i·(2+i)·(2−i)·(3−2i) (2+i)·(2−i)·(3+2i) (2+i)2·(3−2i) i·(2−i)2·(3−2i) |
| 328 | 2+18i 18+2i |
−i·(1+i)3·(5+4i) −i·(1+i)3·(5−4i) |
| 333 | 3+18i 18+3i |
i·3·(6−i) 3·(6+i) |
| 337 | 9+16i 16+9i |
(p) (p) |
| 338 | 7+17i 13+13i 17+7i |
i·(1+i)·(3−2i)2 (1+i)·(3+2i)·(3−2i) −i·(1+i)·(3+2i)2 |
| 340 | 4+18i 12+14i 14+12i 18+4i |
(1+i)2·(2−i)·(4+i) (1+i)2·(2−i)·(4−i) −i·(1+i)2·(2+i)·(4+i) −i·(1+i)2·(2+i)·(4−i) |
| 346 | 11+15i 15+11i |
(1+i)·(13+2i) (1+i)·(13−2i) |
| 349 | 5+18i 18+5i |
(p) (p) |
| 353 | 8+17i 17+8i |
(p) (p) |
| 356 | 10+16i 16+10i |
(1+i)2·(8−5i) −i·(1+i)2·(8+5i) |
| 360 | 6+18i 18+6i |
−i·(1+i)3·(2+i)·3 −i·(1+i)3·(2−i)·3 |
| 361 | 19 | (p) |
| 362 | 1+19i 19+i |
(1+i)·(10+9i) (1+i)·(10−9i) |
| 365 | 2+19i 13+14i 14+13i 19+2i |
i·(2−i)·(8+3i) (2+i)·(8+3i) i·(2−i)·(8−3i) (2+i)·(8−3i) |
| 369 | 12+15i 15+12i |
i·3·(5−4i) 3·(5+4i) |
| 370 | 3+19i 9+17i 17+9i 19+3i |
(1+i)·(2+i)·(6+i) (1+i)·(2+i)·(6−i) (1+i)·(2−i)·(6+i) (1+i)·(2−i)·(6−i) |
| 373 | 7+18i 18+7i |
(p) (p) |
| 377 | 4+19i 11+16i 16+11i 19+4i |
i·(3−2i)·(5+2i) (3+2i)·(5+2i) i·(3−2i)·(5−2i) (3+2i)·(5−2i) |
| 386 | 5+19i 19+5i |
(1+i)·(12+7i) (1+i)·(12−7i) |
| 388 | 8+18i 18+8i |
(1+i)2·(9−4i) −i·(1+i)2·(9+4i) |
| 389 | 10+17i 17+10i |
(p) (p) |
| 392 | 14+14i | −i·(1+i)3·7 |
| 394 | 13+15i 15+13i |
(1+i)·(14+i) (1+i)·(14−i) |
| 397 | 6+19i 19+6i |
(p) (p) |
| 400 | 12+16i 16+12i 20 |
−(1+i)4·(2+i)2 −i·(1+i)4·(2−i)2 −(1+i)4·(2+i)·(2−i) |
| 401 | 1+20i 20+i |
(p) (p) |
| 404 | 2+20i 20+2i |
(1+i)2·(10−i) −i·(1+i)2·(10+i) |
| 405 | 9+18i 18+9i |
i·(2−i)·32 (2+i)·32 |
| 409 | 3+20i 20+3i |
(p) (p) |
| 410 | 7+19i 11+17i 17+11i 19+7i |
i·(1+i)·(2−i)·(5−4i) (1+i)·(2−i)·(5+4i) (1+i)·(2+i)·(5−4i) −i·(1+i)·(2+i)·(5+4i) |
| 416 | 4+20i 20+4i |
−(1+i)5·(3+2i) −(1+i)5·(3−2i) |
| 421 | 14+15i 15+14i |
(p) (p) |
| 424 | 10+18i 18+10i |
−i·(1+i)3·(7+2i) −i·(1+i)3·(7−2i) |
| 425 | 5+20i 8+19i 13+16i 16+13i 19+8i 20+5i |
i·(2+i)·(2−i)·(4−i) (2+i)2·(4+i) i·(2−i)2·(4+i) (2+i)2·(4−i) i·(2−i)2·(4−i) (2+i)·(2−i)·(4+i) |
| 433 | 12+17i 17+12i |
(p) (p) |
| 436 | 6+20i 20+6i |
(1+i)2·(10−3i) −i·(1+i)2·(10+3i) |
| 441 | 21 | 3·7 |
| 442 | 1+21i 9+19i 19+9i 21+i |
i·(1+i)·(3−2i)·(4−i) (1+i)·(3+2i)·(4−i) (1+i)·(3−2i)·(4+i) −i·(1+i)·(3+2i)·(4+i) |
| 445 | 2+21i 11+18i 18+11i 21+2i |
i·(2+i)·(8−5i) (2+i)·(8+5i) i·(2−i)·(8−5i) (2−i)·(8+5i) |
| 449 | 7+20i 20+7i |
(p) (p) |
| 450 | 3+21i 15+15i 21+3i |
i·(1+i)·(2−i)2·3 (1+i)·(2+i)·(2−i)·3 −i·(1+i)·(2+i)2·3 |
| 452 | 14+16i 16+14i |
(1+i)2·(8−7i) −i·(1+i)2·(8+7i) |
| 457 | 4+21i 21+4i |
(p) (p) |
| 458 | 13+17i 17+13i |
(1+i)·(15+2i) (1+i)·(15−2i) |
| 461 | 10+19i 19+10i |
(p) (p) |
| 464 | 8+20i 20+8i |
−i·(1+i)4·(5−2i) −(1+i)4·(5+2i) |
| 466 | 5+21i 21+5i |
(1+i)·(13+8i) (1+i)·(13−8i) |
| 468 | 12+18i 18+12i |
(1+i)2·3·(3−2i) −i·(1+i)2·3·(3+2i) |
| 477 | 6+21i 21+6i |
i·3·(7−2i) 3·(7+2i) |
| 481 | 9+20i 15+16i 16+15i 20+9i |
i·(3−2i)·(6+i) i·(3−2i)·(6−i) (3+2i)·(6+i) (3+2i)·(6−i) |
| 482 | 11+19i 19+11i |
(1+i)·(15+4i) (1+i)·(15−4i) |
| 484 | 22 | −i·(1+i)2·11 |
| 485 | 1+22i 14+17i 17+14i 22+i |
i·(2−i)·(9+4i) (2+i)·(9+4i) i·(2−i)·(9−4i) (2+i)·(9−4i) |
| 488 | 2+22i 22+2i |
−i·(1+i)3·(6+5i) −i·(1+i)3·(6−5i) |
| 490 | 7+21i 21+7i |
(1+i)·(2+i)·7 (1+i)·(2−i)·7 |
| 493 | 3+22i 13+18i 18+13i 22+3i |
i·(4+i)·(5−2i) i·(4−i)·(5−2i) (4+i)·(5+2i) (4−i)·(5+2i) |
| 500 | 4+22i 10+20i 20+10i 22+4i |
−i·(1+i)2·(2+i)3 (1+i)2·(2+i)·(2−i)2 −i·(1+i)2·(2+i)2·(2−i) (1+i)2·(2−i)3 |
| norm | integer | factors |
|---|---|---|
| 505 | 8+21i 12+19i 19+12i 21+8i |
i·(2−i)·(10+i) i·(2−i)·(10−i) (2+i)·(10+i) (2+i)·(10−i) |
| 509 | 5+22i 22+5i |
(p) (p) |
| 512 | 16+16i | (1+i)9 |
| 514 | 15+17i 17+15i |
(1+i)·(16+i) (1+i)·(16−i) |
| 520 | 6+22i 14+18i 18+14i 22+6i |
(1+i)3·(2−i)·(3−2i) −i·(1+i)3·(2−i)·(3+2i) −i·(1+i)3·(2+i)·(3−2i) −(1+i)3·(2+i)·(3+2i) |
| 521 | 11+20i 20+11i |
(p) (p) |
| 522 | 9+21i 21+9i |
(1+i)·3·(5+2i) (1+i)·3·(5−2i) |
| 529 | 23 | (p) |
| 530 | 1+23i 13+19i 19+13i 23+i |
(1+i)·(2+i)·(7+2i) (1+i)·(2+i)·(7−2i) (1+i)·(2−i)·(7+2i) (1+i)·(2−i)·(7−2i) |
| 533 | 2+23i 7+22i 22+7i 23+2i |
i·(3+2i)·(5−4i) (3+2i)·(5+4i) i·(3−2i)·(5−4i) (3−2i)·(5+4i) |
| 538 | 3+23i 23+3i |
(1+i)·(13+10i) (1+i)·(13−10i) |
| 541 | 10+21i 21+10i |
(p) (p) |
| 544 | 12+20i 20+12i |
−(1+i)5·(4+i) −(1+i)5·(4−i) |
| 545 | 4+23i 16+17i 17+16i 23+4i |
i·(2−i)·(10+3i) i·(2−i)·(10−3i) (2+i)·(10+3i) (2+i)·(10−3i) |
| 548 | 8+22i 22+8i |
(1+i)2·(11−4i) −i·(1+i)2·(11+4i) |
| 549 | 15+18i 18+15i |
i·3·(6−5i) 3·(6+5i) |
| 554 | 5+23i 23+5i |
(1+i)·(14+9i) (1+i)·(14−9i) |
| 557 | 14+19i 19+14i |
(p) (p) |
| 562 | 11+21i 21+11i |
(1+i)·(16+5i) (1+i)·(16−5i) |
| 565 | 6+23i 9+22i 22+9i 23+6i |
i·(2+i)·(8−7i) (2+i)·(8+7i) i·(2−i)·(8−7i) (2−i)·(8+7i) |
| 569 | 13+20i 20+13i |
(p) (p) |
| 576 | 24 | i·(1+i)6·3 |
| 577 | 1+24i 24+i |
(p) (p) |
| 578 | 7+23i 17+17i 23+7i |
(1+i)·(4+i)2 (1+i)·(4+i)·(4−i) (1+i)·(4−i)2 |
| 580 | 2+24i 16+18i 18+16i 24+2i |
(1+i)2·(2−i)·(5+2i) −i·(1+i)2·(2+i)·(5+2i) (1+i)2·(2−i)·(5−2i) −i·(1+i)2·(2+i)·(5−2i) |
| 584 | 10+22i 22+10i |
−i·(1+i)3·(8+3i) −i·(1+i)3·(8−3i) |
| 585 | 3+24i 12+21i 21+12i 24+3i |
i·(2+i)·3·(3−2i) (2+i)·3·(3+2i) i·(2−i)·3·(3−2i) (2−i)·3·(3+2i) |
| 586 | 15+19i 19+15i |
(1+i)·(17+2i) (1+i)·(17−2i) |
| 592 | 4+24i 24+4i |
−i·(1+i)4·(6−i) −(1+i)4·(6+i) |
| 593 | 8+23i 23+8i |
(p) (p) |
| 596 | 14+20i 20+14i |
(1+i)2·(10−7i) −i·(1+i)2·(10+7i) |
| 601 | 5+24i 24+5i |
(p) (p) |
| 605 | 11+22i 22+11i |
i·(2−i)·11 (2+i)·11 |
| 610 | 9+23i 13+21i 21+13i 23+9i |
i·(1+i)·(2−i)·(6−5i) (1+i)·(2−i)·(6+5i) (1+i)·(2+i)·(6−5i) −i·(1+i)·(2+i)·(6+5i) |
| 612 | 6+24i 24+6i |
(1+i)2·3·(4−i) −i·(1+i)2·3·(4+i) |
| 613 | 17+18i 18+17i |
(p) (p) |
| 617 | 16+19i 19+16i |
(p) (p) |
| 625 | 7+24i 15+20i 20+15i 24+7i 25 |
−(2−i)4 (2+i)3·(2−i) i·(2+i)·(2−i)3 −i·(2+i)4 (2+i)2·(2−i)2 |
| 626 | 1+25i 25+i |
(1+i)·(13+12i) (1+i)·(13−12i) |
| 628 | 12+22i 22+12i |
(1+i)2·(11−6i) −i·(1+i)2·(11+6i) |
| 629 | 2+25i 10+23i 23+10i 25+2i |
i·(4−i)·(6+i) i·(4−i)·(6−i) (4+i)·(6+i) (4+i)·(6−i) |
| 634 | 3+25i 25+3i |
(1+i)·(14+11i) (1+i)·(14−11i) |
| 637 | 14+21i 21+14i |
i·(3−2i)·7 (3+2i)·7 |
| 640 | 8+24i 24+8i |
i·(1+i)7·(2+i) i·(1+i)7·(2−i) |
| 641 | 4+25i 25+4i |
(p) (p) |
| 648 | 18+18i | −i·(1+i)3·32 |
| 650 | 5+25i 11+23i 17+19i 19+17i 23+11i 25+5i |
(1+i)·(2+i)·(2−i)·(3+2i) (1+i)·(2+i)2·(3−2i) i·(1+i)·(2−i)2·(3−2i) −i·(1+i)·(2+i)2·(3+2i) (1+i)·(2−i)2·(3+2i) (1+i)·(2+i)·(2−i)·(3−2i) |
| 653 | 13+22i 22+13i |
(p) (p) |
| 656 | 16+20i 20+16i |
−i·(1+i)4·(5−4i) −(1+i)4·(5+4i) |
| 657 | 9+24i 24+9i |
i·3·(8−3i) 3·(8+3i) |
| 661 | 6+25i 25+6i |
(p) (p) |
| 666 | 15+21i 21+15i |
(1+i)·3·(6+i) (1+i)·3·(6−i) |
| 673 | 12+23i 23+12i |
(p) (p) |
| 674 | 7+25i 25+7i |
(1+i)·(16+9i) (1+i)·(16−9i) |
| 676 | 10+24i 24+10i 26 |
−i·(1+i)2·(3+2i)2 (1+i)2·(3−2i)2 −i·(1+i)2·(3+2i)·(3−2i) |
| 677 | 1+26i 26+i |
(p) (p) |
| 680 | 2+26i 14+22i 22+14i 26+2i |
−i·(1+i)3·(2+i)·(4+i) −i·(1+i)3·(2+i)·(4−i) −i·(1+i)3·(2−i)·(4+i) −i·(1+i)3·(2−i)·(4−i) |
| 685 | 3+26i 18+19i 19+18i 26+3i |
i·(2−i)·(11+4i) (2+i)·(11+4i) i·(2−i)·(11−4i) (2+i)·(11−4i) |
| 689 | 8+25i 17+20i 20+17i 25+8i |
i·(3−2i)·(7+2i) (3+2i)·(7+2i) i·(3−2i)·(7−2i) (3+2i)·(7−2i) |
| 692 | 4+26i 26+4i |
(1+i)2·(13−2i) −i·(1+i)2·(13+2i) |
| 697 | 11+24i 16+21i 21+16i 24+11i |
i·(4+i)·(5−4i) (4+i)·(5+4i) i·(4−i)·(5−4i) (4−i)·(5+4i) |
| 698 | 13+23i 23+13i |
(1+i)·(18+5i) (1+i)·(18−5i) |
| 701 | 5+26i 26+5i |
(p) (p) |
| 706 | 9+25i 25+9i |
(1+i)·(17+8i) (1+i)·(17−8i) |
| 709 | 15+22i 22+15i |
(p) (p) |
| 712 | 6+26i 26+6i |
−i·(1+i)3·(8+5i) −i·(1+i)3·(8−5i) |
| 720 | 12+24i 24+12i |
−i·(1+i)4·(2−i)·3 −(1+i)4·(2+i)·3 |
| 722 | 19+19i | (1+i)·19 |
| 724 | 18+20i 20+18i |
(1+i)2·(10−9i) −i·(1+i)2·(10+9i) |
| 725 | 7+26i 10+25i 14+23i 23+14i 25+10i 26+7i |
(2+i)2·(5+2i) i·(2+i)·(2−i)·(5−2i) i·(2−i)2·(5+2i) (2+i)2·(5−2i) (2+i)·(2−i)·(5+2i) i·(2−i)2·(5−2i) |
| 729 | 27 | 33 |
| 730 | 1+27i 17+21i 21+17i 27+i |
i·(1+i)·(2−i)·(8−3i) (1+i)·(2+i)·(8−3i) (1+i)·(2−i)·(8+3i) −i·(1+i)·(2+i)·(8+3i) |
| 733 | 2+27i 27+2i |
(p) (p) |
| 738 | 3+27i 27+3i |
(1+i)·3·(5+4i) (1+i)·3·(5−4i) |
| 740 | 8+26i 16+22i 22+16i 26+8i |
(1+i)2·(2−i)·(6+i) (1+i)2·(2−i)·(6−i) −i·(1+i)2·(2+i)·(6+i) −i·(1+i)2·(2+i)·(6−i) |
| 745 | 4+27i 13+24i 24+13i 27+4i |
i·(2+i)·(10−7i) (2+i)·(10+7i) i·(2−i)·(10−7i) (2−i)·(10+7i) |
| 746 | 11+25i 25+11i |
(1+i)·(18+7i) (1+i)·(18−7i) |
| norm | integer | factors |
|---|---|---|
| 754 | 5+27i 15+23i 23+15i 27+5i |
i·(1+i)·(3−2i)·(5−2i) (1+i)·(3+2i)·(5−2i) (1+i)·(3−2i)·(5+2i) −i·(1+i)·(3+2i)·(5+2i) |
| 757 | 9+26i 26+9i |
(p) (p) |
| 761 | 19+20i 20+19i |
(p) (p) |
| 765 | 6+27i 18+21i 21+18i 27+6i |
i·(2−i)·3·(4+i) i·(2−i)·3·(4−i) (2+i)·3·(4+i) (2+i)·3·(4−i) |
| 769 | 12+25i 25+12i |
(p) (p) |
| 772 | 14+24i 24+14i |
(1+i)2·(12−7i) −i·(1+i)2·(12+7i) |
| 773 | 17+22i 22+17i |
(p) (p) |
| 776 | 10+26i 26+10i |
−i·(1+i)3·(9+4i) −i·(1+i)3·(9−4i) |
| 778 | 7+27i 27+7i |
(1+i)·(17+10i) (1+i)·(17−10i) |
| 784 | 28 | −(1+i)4·7 |
| 785 | 1+28i 16+23i 23+16i 28+i |
i·(2+i)·(11−6i) (2+i)·(11+6i) i·(2−i)·(11−6i) (2−i)·(11+6i) |
| 788 | 2+28i 28+2i |
(1+i)2·(14−i) −i·(1+i)2·(14+i) |
| 793 | 3+28i 8+27i 27+8i 28+3i |
i·(3+2i)·(6−5i) (3+2i)·(6+5i) i·(3−2i)·(6−5i) (3−2i)·(6+5i) |
| 794 | 13+25i 25+13i |
(1+i)·(19+6i) (1+i)·(19−6i) |
| 797 | 11+26i 26+11i |
(p) (p) |
| 800 | 4+28i 20+20i 28+4i |
−i·(1+i)5·(2−i)2 −(1+i)5·(2+i)·(2−i) i·(1+i)5·(2+i)2 |
| 801 | 15+24i 24+15i |
i·3·(8−5i) 3·(8+5i) |
| 802 | 19+21i 21+19i |
(1+i)·(20+i) (1+i)·(20−i) |
| 808 | 18+22i 22+18i |
−i·(1+i)3·(10+i) −i·(1+i)3·(10−i) |
| 809 | 5+28i 28+5i |
(p) (p) |
| 810 | 9+27i 27+9i |
(1+i)·(2+i)·32 (1+i)·(2−i)·32 |
| 818 | 17+23i 23+17i |
(1+i)·(20+3i) (1+i)·(20−3i) |
| 820 | 6+28i 12+26i 26+12i 28+6i |
(1+i)2·(2+i)·(5−4i) −i·(1+i)2·(2+i)·(5+4i) (1+i)2·(2−i)·(5−4i) −i·(1+i)2·(2−i)·(5+4i) |
| 821 | 14+25i 25+14i |
(p) (p) |
| 829 | 10+27i 27+10i |
(p) (p) |
| 832 | 16+24i 24+16i |
−(1+i)6·(3−2i) i·(1+i)6·(3+2i) |
| 833 | 7+28i 28+7i |
i·(4−i)·7 (4+i)·7 |
| 841 | 20+21i 21+20i 29 |
i·(5−2i)2 (5+2i)2 (5+2i)·(5−2i) |
| 842 | 1+29i 29+i |
(1+i)·(15+14i) (1+i)·(15−14i) |
| 845 | 2+29i 13+26i 19+22i 22+19i 26+13i 29+2i |
−(2−i)·(3−2i)2 i·(2−i)·(3+2i)·(3−2i) i·(2+i)·(3−2i)2 (2−i)·(3+2i)2 (2+i)·(3+2i)·(3−2i) −i·(2+i)·(3+2i)2 |
| 848 | 8+28i 28+8i |
−i·(1+i)4·(7−2i) −(1+i)4·(7+2i) |
| 850 | 3+29i 11+27i 15+25i 25+15i 27+11i 29+3i |
(1+i)·(2+i)2·(4−i) i·(1+i)·(2−i)2·(4−i) (1+i)·(2+i)·(2−i)·(4+i) (1+i)·(2+i)·(2−i)·(4−i) −i·(1+i)·(2+i)2·(4+i) (1+i)·(2−i)2·(4+i) |
| 853 | 18+23i 23+18i |
(p) (p) |
| 857 | 4+29i 29+4i |
(p) (p) |
| 865 | 9+28i 17+24i 24+17i 28+9i |
i·(2−i)·(13+2i) i·(2−i)·(13−2i) (2+i)·(13+2i) (2+i)·(13−2i) |
| 866 | 5+29i 29+5i |
(1+i)·(17+12i) (1+i)·(17−12i) |
| 872 | 14+26i 26+14i |
−i·(1+i)3·(10+3i) −i·(1+i)3·(10−3i) |
| 873 | 12+27i 27+12i |
i·3·(9−4i) 3·(9+4i) |
| 877 | 6+29i 29+6i |
(p) (p) |
| 881 | 16+25i 25+16i |
(p) (p) |
| 882 | 21+21i | (1+i)·3·7 |
| 884 | 10+28i 20+22i 22+20i 28+10i |
(1+i)2·(3−2i)·(4+i) −i·(1+i)2·(3+2i)·(4+i) (1+i)2·(3−2i)·(4−i) −i·(1+i)2·(3+2i)·(4−i) |
| 890 | 7+29i 19+23i 23+19i 29+7i |
i·(1+i)·(2−i)·(8−5i) (1+i)·(2−i)·(8+5i) (1+i)·(2+i)·(8−5i) −i·(1+i)·(2+i)·(8+5i) |
| 898 | 13+27i 27+13i |
(1+i)·(20+7i) (1+i)·(20−7i) |
| 900 | 18+24i 24+18i 30 |
−i·(1+i)2·(2+i)2·3 (1+i)2·(2−i)2·3 −i·(1+i)2·(2+i)·(2−i)·3 |
| 901 | 1+30i 15+26i 26+15i 30+i |
i·(4+i)·(7−2i) i·(4−i)·(7−2i) (4+i)·(7+2i) (4−i)·(7+2i) |
| 904 | 2+30i 30+2i |
−i·(1+i)3·(8+7i) −i·(1+i)3·(8−7i) |
| 905 | 8+29i 11+28i 28+11i 29+8i |
i·(2+i)·(10−9i) (2+i)·(10+9i) i·(2−i)·(10−9i) (2−i)·(10+9i) |
| 909 | 3+30i 30+3i |
i·3·(10−i) 3·(10+i) |
| 914 | 17+25i 25+17i |
(1+i)·(21+4i) (1+i)·(21−4i) |
| 916 | 4+30i 30+4i |
(1+i)2·(15−2i) −i·(1+i)2·(15+2i) |
| 922 | 9+29i 29+9i |
(1+i)·(19+10i) (1+i)·(19−10i) |
| 925 | 5+30i 14+27i 21+22i 22+21i 27+14i 30+5i |
i·(2+i)·(2−i)·(6−i) (2+i)2·(6+i) i·(2−i)2·(6+i) (2+i)2·(6−i) i·(2−i)2·(6−i) (2+i)·(2−i)·(6+i) |
| 928 | 12+28i 28+12i |
−(1+i)5·(5+2i) −(1+i)5·(5−2i) |
| 929 | 20+23i 23+20i |
(p) (p) |
| 932 | 16+26i 26+16i |
(1+i)2·(13−8i) −i·(1+i)2·(13+8i) |
| 936 | 6+30i 30+6i |
−i·(1+i)3·3·(3+2i) −i·(1+i)3·3·(3−2i) |
| 937 | 19+24i 24+19i |
(p) (p) |
| 941 | 10+29i 29+10i |
(p) (p) |
| 949 | 7+30i 18+25i 25+18i 30+7i |
i·(3−2i)·(8+3i) (3+2i)·(8+3i) i·(3−2i)·(8−3i) (3+2i)·(8−3i) |
| 953 | 13+28i 28+13i |
(p) (p) |
| 954 | 15+27i 27+15i |
(1+i)·3·(7+2i) (1+i)·3·(7−2i) |
| 961 | 31 | (p) |
| 962 | 1+31i 11+29i 29+11i 31+i |
(1+i)·(3+2i)·(6+i) (1+i)·(3+2i)·(6−i) (1+i)·(3−2i)·(6+i) (1+i)·(3−2i)·(6−i) |
| 964 | 8+30i 30+8i |
(1+i)2·(15−4i) −i·(1+i)2·(15+4i) |
| 965 | 2+31i 17+26i 26+17i 31+2i |
i·(2+i)·(12−7i) (2+i)·(12+7i) i·(2−i)·(12−7i) (2−i)·(12+7i) |
| 968 | 22+22i | −i·(1+i)3·11 |
| 970 | 3+31i 21+23i 23+21i 31+3i |
i·(1+i)·(2−i)·(9−4i) (1+i)·(2+i)·(9−4i) (1+i)·(2−i)·(9+4i) −i·(1+i)·(2+i)·(9+4i) |
| 976 | 20+24i 24+20i |
−i·(1+i)4·(6−5i) −(1+i)4·(6+5i) |
| 977 | 4+31i 31+4i |
(p) (p) |
| 980 | 14+28i 28+14i |
(1+i)2·(2−i)·7 −i·(1+i)2·(2+i)·7 |
| 981 | 9+30i 30+9i |
i·3·(10−3i) 3·(10+3i) |
| 985 | 12+29i 16+27i 27+16i 29+12i |
i·(2−i)·(14+i) i·(2−i)·(14−i) (2+i)·(14+i) (2+i)·(14−i) |
| 986 | 5+31i 19+25i 25+19i 31+5i |
(1+i)·(4+i)·(5+2i) (1+i)·(4−i)·(5+2i) (1+i)·(4+i)·(5−2i) (1+i)·(4−i)·(5−2i) |
| 997 | 6+31i 31+6i |
(p) (p) |
| 1000 | 10+30i 18+26i 26+18i 30+10i |
−i·(1+i)3·(2+i)2·(2−i) (1+i)3·(2−i)3 −(1+i)3·(2+i)3 −i·(1+i)3·(2+i)·(2−i)2 |
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.