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Table of Gaussian integer factorizations

Mathematical table

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 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 x2 + y2 . 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

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

References

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This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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