Table of Gaussian integer factorizations

Short description: Mathematical table

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.

## Contents

* 1 Conventions
* 2 Factorizations
* 3 See also
* 4 References
* 5 External links

## 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

Template:Table TOC

Norm 1–250  Norm | Integer | Factorization

|
|

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 251–500  Norm | Integer | Factorization

|
|

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 501–750  Norm | Integer | Factorization

|
|

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 751–1000  Norm | Integer | Factorization

|
|

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

* Gaussian integer
* Table of divisors
* Integer factorization

## 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.
* Gethner, Ellen; Wagner, Stan; Wick, Brian (1998). "A stroll through the Gaussian primes". Amer. Math. Monthly 105 (4): 327–337. doi:10.2307/2589708.
* 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.

## External links

* Weisstein, Eric W.. "Gaussian prime". http://mathworld.wolfram.com/GaussianPrime.html.
* Weisstein, Eric W.. "Prime Factorization". http://mathworld.wolfram.com/PrimeFactorization.html.
* OEIS: Gaussian Primes

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