Tabela rozkładu liczb całkowitych Gaussa na czynniki pierwsze

Liczba całkowita Gaussa jest zerem, jedną z czterech jedności $(\pm 1,\pm i),$ liczbą pierwszą Gaussa lub liczbą złożoną. Artykuł stanowi tabela liczb całkowitych Gaussa $x+yi,$ których norma nie przekracza 1000 i ich rozkład na czynniki pierwsze. Liczby pierwsze Gaussa oznaczone są w tabeli etykietą (p). Rozkład na czynniki obejmuje również opcjonalną jedność pomnożoną przez potęgi liczb pierwszych Gaussa dla większej przejrzystości zapisu.

Zwrócić uwagę należy na istnienie wymiernych liczb pierwszych (tzn. liczb pierwszych ze zbioru liczb całkowitych), które nie są liczbami pierwszymi Gaussa. Przykładem jest wymierna liczba pierwsza 5, która w zbiorze $\mathbb {Z} [i]$ jest rozkładalna na czynniki: $5=(2-i)(2+i),$ nie jest w związku z tym liczbą pierwszą Gaussa.

Konwencje

Druga kolumna tabeli zawiera tylko liczby z pierwszej ćwiartki, tzn. liczby, których część rzeczywista $x$ jest dodatnia, a część urojona $y$ jest nieujemna. Tabelę można było jeszcze bardziej uprościć sprowadzając liczby do tych z pierwszej oktanty płaszczyzny zespolonej przy użyciu obserwacji, że $y+xi=i(x-yi).$

Rozkłady na czynniki nie są wyjątkowe w tym sensie, że jednostka może zostać wchłonięta przez dowolny inny czynnik z wykładnikiem równym jeden. Na przykład $4+2i=-i(1+i)^{2}(2+i)$ można zapisać również jako $4+2i=(1+i)^{2}(1-2i).$ Ta niejednoznaczność została rozwiązana zgodnie z następującą konwencją: czynniki są liczbami pierwszymi z prawej półpłaszczyzny zespolonej o wartości bezwzględnej części rzeczywistej większej lub równej wartości bezwzględnej części urojonej.

Wpisy są sortowane według rosnącej normy $x^{2}+y^{2}$ (ciąg 001481 w OEIS). Tabela jest kompletna do maksymalnej normy na końcu tabeli w tym sensie, że każda liczba złożona lub pierwsza z pierwszej ćwiartki pojawia się w drugiej kolumnie.

Liczby pierwsze Gaussa występują tylko dla pewnego podzbioru norm (ciąg A055025 w OEIS). Jest to bardziej przejrzysta wersja dwóch innych ciągów: (ciąg A103431 w OEIS), (ciąg A103432 w OEIS).

Rozkład na czynniki

Norma	Liczba	Czynniki
$2$	$1+i$	(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)
$18$	$3+3i$	$(1+i)\cdot 3$
$20$	$2+4i$ $4+2i$	$(1+i)^{2}(2-i)$ $-i(1+i)^{2}(2+i)$
$25$	$3+4i$ $4+3i$ $5$	$(2+i)^{2}$ $i(2-i)^{2}$ $(2+i)(2-i)$
$26$	$1+5i$ $5+i$	$(1+i)(3+2i)$ $(1+i)(3-2i)$
$29$	$2+5i$ $5+2i$	(p) (p)
$32$	$4+4i$	$-(1+i)^{5}$
$34$	$3+5i$ $5+3i$	$(1+i)(4+i)$ $(1+i)(4-i)$
$36$	$6$	$-i(1+i)^{2}\cdot 3$
$37$	$1+6i$ $6+i$	(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)
$45$	$3+6i$ $6+3i$	$i(2-i)\cdot 3$ $(2+i)\cdot 3$
$49$	$7$	(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)
$64$	$8$	$i(1+i)^{6}$
$65$	$1+8i$ $4+7i$ $7+4i$ $8+i$	$i(2+i)(3-2i)$ $(2+i)(3+2i)$ $i(2-i)(3-2i)$ $(2-i)(3+2i)$
$68$	$2+8i$ $8+2i$	$(1+i)^{2}(4-i)$ $-i(1+i)^{2}(4+i)$
$72$	$6+6i$	$-i(1+i)^{3}\cdot 3$
$73$	$3+8i$ $8+3i$	(p) (p)
$74$	$5+7i$ $7+5i$	$(1+i)(6+i)$ $(1+i)(6-i)$
$80$	$4+8i$ $8+4i$	$-i(1+i)^{4}(2-i)$ $-(1+i)^{4}(2+i)$
$81$	$9$	$3^{2}$
$82$	$1+9i$ $9+i$	$(1+i)(5+4i)$ $(1+i)(5-4i)$
$85$	$2+9i$ $6+7i$ $7+6i$ $9+2i$	$i(2-i)(4+i)$ $i(2-i)(4-i)$ $(2+i)(4+i)$ $(2+i)(4-i)$
$89$	$5+8i$ $8+5i$	(p) (p)
$90$	$3+9i$ $9+3i$	$(1+i)(2+i)\cdot 3$ $(1+i)(2-i)\cdot 3$
$97$	$4+9i$ $9+4i$	(p) (p)
$98$	$7+7i$	$(1+i)\cdot 7$
$100$	$6+8i$ $8+6i$ $10$	$-i(1+i)^{2}(2+i)^{2}$ $(1+i)^{2}(2-i)^{2}$ $-i(1+i)^{2}(2+i)(2-i)$
$101$	$1+10i$ $10+i$	(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)
$116$	$4+10i$ $10+4i$	$(1+i)^{2}(5-2i)$ $-i(1+i)^{2}(5+2i)$
$117$	$6+9i$ $9+6i$	$i\cdot 3\cdot (3-2i)$ $3\cdot (3+2i)$
$121$	$11$	(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)
$144$	$12$	$-(1+i)^{4}\cdot 3$
$145$	$1+12i$ $8+9i$ $9+8i$ $12+i$	$i(2-i)(5+2i)$ $(2+i)(5+2i)$ $i(2-i)(5-2i)$ $(2+i)(5-2i)$
$146$	$5+11i$ $11+5i$	$(1+i)(8+3i)$ $(1+i)(8-3i)$
$148$	$2+12i$ $12+2i$	$(1+i)^{2}(6-i)$ $-i(1+i)^{2}(6+i)$
$149$	$7+10i$ $10+7i$	(p) (p)
$153$	$3+12i$ $12+3i$	$i\cdot 3\cdot (4-i)$ $3\cdot (4+i)$
$157$	$6+11i$ $11+6i$	(p) (p)
$160$	$4+12i$ $12+4i$	$-(1+i)^{5}(2+i)$ $-(1+i)^{5}(2-i)$
$162$	$9+9i$	$(1+i)\cdot 3^{2}$
$164$	$8+10i$ $10+8i$	$(1+i)^{2}(5-4i)$ $-i(1+i)^{2}(5+4i)$
$169$	$5+12i$ $12+5i$ $13$	$(3+2i)^{2}$ $i(3-2i)^{2}$ $(3+2i)(3-2i)$
$170$	$1+13i$ $7+11i$ $11+7i$ $13+i$	$(1+i)(2+i)(4+i)$ $(1+i)(2+i)(4-i)$ $(1+i)(2-i)(4+i)$ $(1+i)(2-i)(4-i)$
$173$	$2+13i$ $13+2i$	(p) (p)
$178$	$3+13i$ $13+3i$	$(1+i)(8+5i)$ $(1+i)(8-5i)$
$180$	$6+12i$ $12+6i$	$(1+i)^{2}(2-i)\cdot 3$ $-i(1+i)^{2}(2+i)\cdot 3$
$181$	$9+10i$ $10+9i$	(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)
$194$	$5+13i$ $13+5i$	$(1+i)(9+4i)$ $(1+i)(9-4i)$
$196$	$14$	$-i(1+i)^{2}\cdot 7$
$197$	$1+14i$ $14+i$	(p) (p)
$200$	$2+14i$ $10+10i$ $14+2i$	$(1+i)^{3}(2-i)^{2}$ $-i(1+i)^{3}(2+i)(2-i)$ $-(1+i)^{3}(2+i)^{2}$
$202$	$9+11i$ $11+9i$	$(1+i)(10+i)$ $(1+i)(10-i)$
$205$	$3+14i$ $6+13i$ $13+6i$ $14+3i$	$i(2+i)(5-4i)$ $(2+i)(5+4i)$ $i(2-i)(5-4i)$ $(2-i)(5+4i)$
$208$	$8+12i$ $12+8i$	$-i(1+i)^{4}(3-2i)$ $-(1+i)^{4}(3+2i)$
$212$	$4+14i$ $14+4i$	$(1+i)^{2}(7-2i)$ $-i(1+i)^{2}(7+2i)$
$218$	$7+13i$ $13+7i$	$(1+i)(10+3i)$ $(1+i)(10-3i)$
$221$	$5+14i$ $10+11i$ $11+10i$ $14+5i$	$i(3-2i)(4+i)$ $(3+2i)(4+i)$ $i(3-2i)(4-i)$ $(3+2i)(4-i)$
$225$	$9+12i$ $12+9i$ $15$	$(2+i)^{2}\cdot 3$ $i(2-i)^{2}\cdot 3$ $(2+i)(2-i)\cdot 3$
$226$	$1+15i$ $15+i$	$(1+i)(8+7i)$ $(1+i)(8-7i)$
$229$	$2+15i$ $15+2i$	(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)
$234$	$3+15i$ $15+3i$	$(1+i)\cdot 3\cdot (3+2i)$ $(1+i)\cdot 3\cdot (3-2i)$
$241$	$4+15i$ $15+4i$	(p) (p)
$242$	$11+11i$	$(1+i)\cdot 11$
$244$	$10+12i$ $12+10i$	$(1+i)^{2}(6-5i)$ $-i(1+i)^{2}(6+5i)$
$245$	$7+14i$ $14+7i$	$i(2-i)\cdot 7$ $(2+i)\cdot 7$
$250$	$5+15i$ $9+13i$ $13+9i$ $15+5i$	$(1+i)(2+i)^{2}(2-i)$ $i(1+i)(2-i)^{3}$ $-i(1+i)(2+i)^{3}$ $(1+i)(2+i)(2-i)^{2}$

Norma	Liczba	Czynniki
$256$	$16$	$(1+i)^{8}$
$257$	$1+16i$ $16+i$	(p) (p)
$260$	$2+16i$ $8+14i$ $14+8i$ $16+2i$	$(1+i)^{2}(2+i)(3-2i)$ $-i(1+i)^{2}(2+i)(3+2i)$ $(1+i)^{2}(2-i)(3-2i)$ $-i(1+i)^{2}(2-i)(3+2i)$
$261$	$6+15i$ $15+6i$	$i\cdot 3\cdot (5-2i)$ $3\cdot (5+2i)$
$265$	$3+16i$ $11+12i$ $12+11i$ $16+3i$	$i(2-i)(7+2i)$ $i(2-i)(7-2i)$ $(2+i)(7+2i)$ $(2+i)(7-2i)$
$269$	$10+13i$ $13+10i$	(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)
$288$	$12+12i$	$-(1+i)^{5}\cdot 3$
$289$	$8+15i$ $15+8i$ $17$	$i(4-i)^{2}$ $(4+i)^{2}$ $(4+i)(4-i)$
$290$	$1+17i$ $11+13i$ $13+11i$ $17+i$	$i(1+i)(2-i)(5-2i)$ $(1+i)(2+i)(5-2i)$ $(1+i)(2-i)(5+2i)$ $-i(1+i)(2+i)(5+2i)$
$292$	$6+16i$ $16+6i$	$(1+i)^{2}(8-3i)$ $-i(1+i)^{2}(8+3i)$
$293$	$2+17i$ $17+2i$	(p) (p)
$296$	$10+14i$ $14+10i$	$-i(1+i)^{3}(6+i)$ $-i(1+i)^{3}(6-i)$
$298$	$3+17i$ $17+3i$	$(1+i)(10+7i)$ $(1+i)(10-7i)$
$305$	$4+17i$ $7+16i$ $16+7i$ $17+4i$	$i(2+i)(6-5i)$ $(2+i)(6+5i)$ $i(2-i)(6-5i)$ $(2-i)(6+5i)$
$306$	$9+15i$ $15+9i$	$(1+i)\cdot 3\cdot (4+i)$ $(1+i)\cdot 3\cdot (4-i)$
$313$	$12+13i$ $13+12i$	(p) (p)
$314$	$5+17i$ $17+5i$	$(1+i)(11+6i)$ $(1+i)(11-6i)$
$317$	$11+14i$ $14+11i$	(p) (p)
$320$	$8+16i$ $16+8i$	$-(1+i)^{6}(2-i)$ $i(1+i)^{6}(2+i)$
$324$	$18$	$-i(1+i)^{2}\cdot 3^{2}$
$325$	$1+18i$ $6+17i$ $10+15i$ $15+10i$ $17+6i$ $18+i$	$(2+i)^{2}(3+2i)$ $i(2-i)^{2}(3+2i)$ $i(2+i)(2-i)(3-2i)$ $(2+i)(2-i)(3+2i)$ $(2+i)^{2}(3-2i)$ $i(2-i)^{2}(3-2i)$
$328$	$2+18i$ $18+2i$	$-i(1+i)^{3}(5+4i)$ $-i(1+i)^{3}(5-4i)$
$333$	$3+18i$ $18+3i$	$i\cdot 3\cdot (6-i)$ $3\cdot (6+i)$
$337$	$9+16i$ $16+9i$	(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)
$356$	$10+16i$ $16+10i$	$(1+i)^{2}(8-5i)$ $-i(1+i)^{2}(8+5i)$
$360$	$6+18i$ $18+6i$	$-i(1+i)^{3}(2+i)\cdot 3$ $-i(1+i)^{3}(2-i)\cdot 3$
$361$	$19$	(p)
$362$	$1+19i$ $19+i$	$(1+i)(10+9i)$ $(1+i)(10-9i)$
$365$	$2+19i$ $13+14i$ $14+13i$ $19+2i$	$i(2-i)(8+3i)$ $(2+i)(8+3i)$ $i(2-i)(8-3i)$ $(2+i)(8-3i)$
$369$	$12+15i$ $15+12i$	$i\cdot 3\cdot (5-4i)$ $3\cdot (5+4i)$
$370$	$3+19i$ $9+17i$ $17+9i$ $19+3i$	$(1+i)(2+i)(6+i)$ $(1+i)(2+i)(6-i)$ $(1+i)(2-i)(6+i)$ $(1+i)(2-i)(6-i)$
$373$	$7+18i$ $18+7i$	(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)
$392$	$14+14i$	$-i(1+i)^{3}\cdot 7$
$394$	$13+15i$ $15+13i$	$(1+i)(14+i)$ $(1+i)(14-i)$
$397$	$6+19i$ $19+6i$	(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)
$404$	$2+20i$ $20+2i$	$(1+i)^{2}(10-i)$ $-i(1+i)^{2}(10+i)$
$405$	$9+18i$ $18+9i$	$i(2-i)\cdot 3^{2}$ $(2+i)\cdot 3^{2}$
$409$	$3+20i$ $20+3i$	(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)
$436$	$6+20i$ $20+6i$	$(1+i)^{2}(10-3i)$ $-i(1+i)^{2}(10+3i)$
$441$	$21$	$3\cdot 7$
$442$	$1+21i$ $9+19i$ $19+9i$ $21+i$	$i(1+i)(3-2i)(4-i)$ $(1+i)(3+2i)(4-i)$ $(1+i)(3-2i)(4+i)$ $-i(1+i)(3+2i)(4+i)$
$445$	$2+21i$ $11+18i$ $18+11i$ $21+2i$	$i(2+i)(8-5i)$ $(2+i)(8+5i)$ $i(2-i)(8-5i)$ $(2-i)(8+5i)$
$449$	$7+20i$ $20+7i$	(p) (p)
$450$	$3+21i$ $15+15i$ $21+3i$	$i(1+i)(2-i)^{2}\cdot 3$ $(1+i)(2+i)(2-i)\cdot 3$ $-i(1+i)(2+i)^{2}\cdot 3$
$452$	$14+16i$ $16+14i$	$(1+i)^{2}(8-7i)$ $-i(1+i)^{2}(8+7i)$
$457$	$4+21i$ $21+4i$	(p) (p)
$458$	$13+17i$ $17+13i$	$(1+i)(15+2i)$ $(1+i)(15-2i)$
$461$	$10+19i$ $19+10i$	(p) (p)
$464$	$8+20i$ $20+8i$	$-i(1+i)^{4}(5-2i)$ $-(1+i)^{4}(5+2i)$
$466$	$5+21i$ $21+5i$	$(1+i)(13+8i)$ $(1+i)(13-8i)$
$468$	$12+18i$ $18+12i$	$(1+i)^{2}\cdot 3\cdot (3-2i)$ $-i(1+i)^{2}\cdot 3\cdot (3+2i)$
$477$	$6+21i$ $21+6i$	$i\cdot 3\cdot (7-2i)$ $3\cdot (7+2i)$
$481$	$9+20i$ $15+16i$ $16+15i$ $20+9i$	$i(3-2i)(6+i)$ $i(3-2i)(6-i)$ $(3+2i)(6+i)$ $(3+2i)(6-i)$
$482$	$11+19i$ $19+11i$	$(1+i)(15+4i)$ $(1+i)(15-4i)$
$484$	$22$	$-i(1+i)^{2}\cdot 11$
$485$	$1+22i$ $14+17i$ $17+14i$ $22+i$	$i(2-i)(9+4i)$ $(2+i)(9+4i)$ $i(2-i)(9-4i)$ $(2+i)(9-4i)$
$488$	$2+22i$ $22+2i$	$-i(1+i)^{3}(6+5i)$ $-i(1+i)^{3}(6-5i)$
$490$	$7+21i$ $21+7i$	$(1+i)(2+i)\cdot 7$ $(1+i)(2-i)\cdot 7$
$493$	$3+22i$ $13+18i$ $18+13i$ $22+3i$	$i(4+i)(5-2i)$ $i(4-i)(5-2i)$ $(4+i)(5+2i)$ $(4-i)(5+2i)$
$500$	$4+22i$ $10+20i$ $20+10i$ $22+4i$	$-i(1+i)^{2}(2+i)^{3}$ $(1+i)^{2}(2+i)(2-i)^{2}$ $-i(1+i)^{2}(2+i)^{2}(2-i)$ $(1+i)^{2}(2-i)^{3}$

Norma	Liczba	Czynniki
$505$	$8+21i$ $12+19i$ $19+12i$ $21+8i$	$i(2-i)(10+i)$ $i(2-i)(10-i)$ $(2+i)(10+i)$ $(2+i)(10-i)$
$509$	$5+22i$ $22+5i$	(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)
$522$	$9+21i$ $21+9i$	$(1+i)\cdot 3\cdot (5+2i)$ $(1+i)\cdot 3\cdot (5-2i)$
$529$	$23$	(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)
$544$	$12+20i$ $20+12i$	$-(1+i)^{5}(4+i)$ $-(1+i)^{5}(4-i)$
$545$	$4+23i$ $16+17i$ $17+16i$ $23+4i$	$i(2-i)(10+3i)$ $i(2-i)(10-3i)$ $(2+i)(10+3i)$ $(2+i)(10-3i)$
$548$	$8+22i$ $22+8i$	$(1+i)^{2}(11-4i)$ $-i(1+i)^{2}(11+4i)$
$549$	$15+18i$ $18+15i$	$i\cdot 3\cdot (6-5i)$ $3\cdot (6+5i)$
$554$	$5+23i$ $23+5i$	$(1+i)(14+9i)$ $(1+i)(14-9i)$
$557$	$14+19i$ $19+14i$	(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)
$576$	$24$	$i(1+i)^{6}\cdot 3$
$577$	$1+24i$ $24+i$	(p) (p)
$578$	$7+23i$ $17+17i$ $23+7i$	$(1+i)(4+i)^{2}$ $(1+i)(4+i)(4-i)$ $(1+i)(4-i)^{2}$
$580$	$2+24i$ $16+18i$ $18+16i$ $24+2i$	$(1+i)^{2}(2-i)(5+2i)$ $-i(1+i)^{2}(2+i)(5+2i)$ $(1+i)^{2}(2-i)(5-2i)$ $-i(1+i)^{2}(2+i)(5-2i)$
$584$	$10+22i$ $22+10i$	$-i(1+i)^{3}(8+3i)$ $-i(1+i)^{3}(8-3i)$
$585$	$3+24i$ $12+21i$ $21+12i$ $24+3i$	$i(2+i)\cdot 3\cdot (3-2i)$ $(2+i)\cdot 3\cdot (3+2i)$ $i(2-i)\cdot 3\cdot (3-2i)$ $(2-i)\cdot 3\cdot (3+2i)$
$586$	$15+19i$ $19+15i$	$(1+i)(17+2i)$ $(1+i)(17-2i)$
$592$	$4+24i$ $24+4i$	$-i(1+i)^{4}(6-i)$ $-(1+i)^{4}(6+i)$
$593$	$8+23i$ $23+8i$	(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)
$605$	$11+22i$ $22+11i$	$i(2-i)\cdot 11$ $(2+i)\cdot 11$
$610$	$9+23i$ $13+21i$ $21+13i$ $23+9i$	$i(1+i)(2-i)(6-5i)$ $(1+i)(2-i)(6+5i)$ $(1+i)(2+i)(6-5i)$ $-i(1+i)(2+i)(6+5i)$
$612$	$6+24i$ $24+6i$	$(1+i)^{2}\cdot 3\cdot (4-i)$ $-i(1+i)^{2}\cdot 3\cdot (4+i)$
$613$	$17+18i$ $18+17i$	(p) (p)
$617$	$16+19i$ $19+16i$	(p) (p)
$625$	$7+24i$ $15+20i$ $20+15i$ $24+7i$ $25$	$-(2-i)^{4}$ $(2+i)^{3}(2-i)$ $i(2+i)(2-i)^{3}$ $-i(2+i)^{4}$ $(2+i)^{2}(2-i)^{2}$
$626$	$1+25i$ $25+i$	$(1+i)(13+12i)$ $(1+i)(13-12i)$
$628$	$12+22i$ $22+12i$	$(1+i)^{2}(11-6i)$ $-i(1+i)^{2}(11+6i)$
$629$	$2+25i$ $10+23i$ $23+10i$ $25+2i$	$i(4-i)(6+i)$ $i(4-i)(6-i)$ $(4+i)(6+i)$ $(4+i)(6-i)$
$634$	$3+25i$ $25+3i$	$(1+i)(14+11i)$ $(1+i)(14-11i)$
$637$	$14+21i$ $21+14i$	$i(3-2i)\cdot 7$ $(3+2i)\cdot 7$
$640$	$8+24i$ $24+8i$	$i(1+i)^{7}(2+i)$ $i(1+i)^{7}(2-i)$
$641$	$4+25i$ $25+4i$	(p) (p)
$648$	$18+18i$	$-i(1+i)^{3}\cdot 3^{2}$
$650$	$5+25i$ $11+23i$ $17+19i$ $19+17i$ $23+11i$ $25+5i$	$(1+i)(2+i)(2-i)(3+2i)$ $(1+i)(2+i)^{2}(3-2i)$ $i(1+i)(2-i)^{2}(3-2i)$ $-i(1+i)(2+i)^{2}(3+2i)$ $(1+i)(2-i)^{2}(3+2i)$ $(1+i)(2+i)(2-i)(3-2i)$
$653$	$13+22i$ $22+13i$	(p) (p)
$656$	$16+20i$ $20+16i$	$-i(1+i)^{4}(5-4i)$ $-(1+i)^{4}(5+4i)$
$657$	$9+24i$ $24+9i$	$i\cdot 3\cdot (8-3i)$ $3\cdot (8+3i)$
$661$	$6+25i$ $25+6i$	(p) (p)
$666$	$15+21i$ $21+15i$	$(1+i)\cdot 3\cdot (6+i)$ $(1+i)\cdot 3\cdot (6-i)$
$673$	$12+23i$ $23+12i$	(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)
$712$	$6+26i$ $26+6i$	$-i(1+i)^{3}(8+5i)$ $-i(1+i)^{3}(8-5i)$
$720$	$12+24i$ $24+12i$	$-i(1+i)^{4}(2-i)\cdot 3$ $-(1+i)^{4}(2+i)\cdot 3$
$722$	$19+19i$	$(1+i)\cdot 19$
$724$	$18+20i$ $20+18i$	$(1+i)^{2}(10-9i)$ $-i(1+i)^{2}(10+9i)$
$725$	$7+26i$ $10+25i$ $14+23i$ $23+14i$ $25+10i$ $26+7i$	$(2+i)^{2}(5+2i)$ $i(2+i)(2-i)(5-2i)$ $i(2-i)^{2}(5+2i)$ $(2+i)^{2}(5-2i)$ $(2+i)(2-i)(5+2i)$ $i(2-i)^{2}(5-2i)$
$729$	$27$	$3^{3}$
$730$	$1+27i$ $17+21i$ $21+17i$ $27+i$	$i(1+i)(2-i)(8-3i)$ $(1+i)(2+i)(8-3i)$ $(1+i)(2-i)(8+3i)$ $-i(1+i)(2+i)(8+3i)$
$733$	$2+27i$ $27+2i$	(p) (p)
$738$	$3+27i$ $27+3i$	$(1+i)\cdot 3\cdot (5+4i)$ $(1+i)\cdot 3\cdot (5-4i)$
$740$	$8+26i$ $16+22i$ $22+16i$ $26+8i$	$(1+i)^{2}(2-i)(6+i)$ $(1+i)^{2}(2-i)(6-i)$ $-i(1+i)^{2}(2+i)(6+i)$ $-i(1+i)^{2}(2+i)(6-i)$
$745$	$4+27i$ $13+24i$ $24+13i$ $27+4i$	$i(2+i)(10-7i)$ $(2+i)(10+7i)$ $i(2-i)(10-7i)$ $(2-i)(10+7i)$
$746$	$11+25i$ $25+11i$	$(1+i)(18+7i)$ $(1+i)(18-7i)$

Norma	Liczba	Czynniki
$754$	$5+27i$ $15+23i$ $23+15i$ $27+5i$	$i(1+i)(3-2i)(5-2i)$ $(1+i)(3+2i)(5-2i)$ $(1+i)(3-2i)(5+2i)$ $-i(1+i)(3+2i)(5+2i)$
$757$	$9+26i$ $26+9i$	(p) (p)
$761$	$19+20i$ $20+19i$	(p) (p)
$765$	$6+27i$ $18+21i$ $21+18i$ $27+6i$	$i(2-i)\cdot 3\cdot (4+i)$ $i(2-i)\cdot 3\cdot (4-i)$ $(2+i)\cdot 3\cdot (4+i)$ $(2+i)\cdot 3\cdot (4-i)$
$769$	$12+25i$ $25+12i$	(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)
$776$	$10+26i$ $26+10i$	$-i(1+i)^{3}(9+4i)$ $-i(1+i)^{3}(9-4i)$
$778$	$7+27i$ $27+7i$	$(1+i)(17+10i)$ $(1+i)(17-10i)$
$784$	$28$	$-(1+i)^{4}\cdot 7$
$785$	$1+28i$ $16+23i$ $23+16i$ $28+i$	$i(2+i)(11-6i)$ $(2+i)(11+6i)$ $i(2-i)(11-6i)$ $(2-i)(11+6i)$
$788$	$2+28i$ $28+2i$	$(1+i)^{2}(14-i)$ $-i(1+i)^{2}(14+i)$
$793$	$3+28i$ $8+27i$ $27+8i$ $28+3i$	$i(3+2i)(6-5i)$ $(3+2i)(6+5i)$ $i(3-2i)(6-5i)$ $(3-2i)(6+5i)$
$794$	$13+25i$ $25+13i$	$(1+i)(19+6i)$ $(1+i)(19-6i)$
$797$	$11+26i$ $26+11i$	(p) (p)
$800$	$4+28i$ $20+20i$ $28+4i$	$-i(1+i)^{5}(2-i)^{2}$ $-(1+i)^{5}(2+i)(2-i)$ $i(1+i)^{5}(2+i)^{2}$
$801$	$15+24i$ $24+15i$	$i\cdot 3\cdot (8-5i)$ $3\cdot (8+5i)$
$802$	$19+21i$ $21+19i$	$(1+i)(20+i)$ $(1+i)(20-i)$
$808$	$18+22i$ $22+18i$	$-i(1+i)^{3}(10+i)$ $-i(1+i)^{3}(10-i)$
$809$	$5+28i$ $28+5i$	(p) (p)
$810$	$9+27i$ $27+9i$	$(1+i)(2+i)\cdot 3^{2}$ $(1+i)(2-i)\cdot 3^{2}$
$818$	$17+23i$ $23+17i$	$(1+i)(20+3i)$ $(1+i)(20-3i)$
$820$	$6+28i$ $12+26i$ $26+12i$ $28+6i$	$(1+i)^{2}(2+i)(5-4i)$ $-i(1+i)^{2}(2+i)(5+4i)$ $(1+i)^{2}(2-i)(5-4i)$ $-i(1+i)^{2}(2-i)(5+4i)$
$821$	$14+25i$ $25+14i$	(p) (p)
$829$	$10+27i$ $27+10i$	(p) (p)
$832$	$16+24i$ $24+16i$	$-(1+i)^{6}(3-2i)$ $i(1+i)^{6}(3+2i)$
$833$	$7+28i$ $28+7i$	$i(4-i)\cdot 7$ $(4+i)\cdot 7$
$841$	$20+21i$ $21+20i$ $29$	$i(5-2i)^{2}$ $(5+2i)^{2}$ $(5+2i)(5-2i)$
$842$	$1+29i$ $29+i$	$(1+i)(15+14i)$ $(1+i)(15-14i)$
$845$	$2+29i$ $13+26i$ $19+22i$ $22+19i$ $26+13i$ $29+2i$	$-(2-i)(3-2i)^{2}$ $i(2-i)(3+2i)(3-2i)$ $i(2+i)(3-2i)^{2}$ $(2-i)(3+2i)^{2}$ $(2+i)(3+2i)(3-2i)$ $-i(2+i)(3+2i)^{2}$
$848$	$8+28i$ $28+8i$	$-i(1+i)^{4}(7-2i)$ $-(1+i)^{4}(7+2i)$
$850$	$3+29i$ $11+27i$ $15+25i$ $25+15i$ $27+11i$ $29+3i$	$(1+i)(2+i)^{2}(4-i)$ $i(1+i)(2-i)^{2}(4-i)$ $(1+i)(2+i)(2-i)(4+i)$ $(1+i)(2+i)(2-i)(4-i)$ $-i(1+i)(2+i)^{2}(4+i)$ $(1+i)(2-i)^{2}(4+i)$
$853$	$18+23i$ $23+18i$	(p) (p)
$857$	$4+29i$ $29+4i$	(p) (p)
$865$	$9+28i$ $17+24i$ $24+17i$ $28+9i$	$i(2-i)(13+2i)$ $i(2-i)(13-2i)$ $(2+i)(13+2i)$ $(2+i)(13-2i)$
$866$	$5+29i$ $29+5i$	$(1+i)(17+12i)$ $(1+i)(17-12i)$
$872$	$14+26i$ $26+14i$	$-i(1+i)^{3}(10+3i)$ $-i(1+i)^{3}(10-3i)$
$873$	$12+27i$ $27+12i$	$i\cdot 3\cdot (9-4i)$ $3\cdot (9+4i)$
$877$	$6+29i$ $29+6i$	(p) (p)
$881$	$16+25i$ $25+16i$	(p) (p)
$882$	$21+21i$	$(1+i)\cdot 3\cdot 7$
$884$	$10+28i$ $20+22i$ $22+20i$ $28+10i$	$(1+i)^{2}(3-2i)(4+i)$ $-i(1+i)^{2}(3+2i)(4+i)$ $(1+i)^{2}(3-2i)(4-i)$ $-i(1+i)^{2}(3+2i)(4-i)$
$890$	$7+29i$ $19+23i$ $23+19i$ $29+7i$	$i(1+i)(2-i)(8-5i)$ $(1+i)(2-i)(8+5i)$ $(1+i)(2+i)(8-5i)$ $-i(1+i)(2+i)(8+5i)$
$898$	$13+27i$ $27+13i$	$(1+i)(20+7i)$ $(1+i)(20-7i)$
$900$	$18+24i$ $24+18i$ $30$	$-i(1+i)^{2}(2+i)^{2}\cdot 3$ $(1+i)^{2}(2-i)^{2}\cdot 3$ $-i(1+i)^{2}(2+i)(2-i)\cdot 3$
$901$	$1+30i$ $15+26i$ $26+15i$ $30+i$	$i(4+i)(7-2i)$ $i(4-i)(7-2i)$ $(4+i)(7+2i)$ $(4-i)(7+2i)$
$904$	$2+30i$ $30+2i$	$-i(1+i)^{3}(8+7i)$ $-i(1+i)^{3}(8-7i)$
$905$	$8+29i$ $11+28i$ $28+11i$ $29+8i$	$i(2+i)(10-9i)$ $(2+i)(10+9i)$ $i(2-i)(10-9i)$ $(2-i)(10+9i)$
$909$	$3+30i$ $30+3i$	$i\cdot 3\cdot (10-i)$ $3\cdot (10+i)$
$914$	$17+25i$ $25+17i$	$(1+i)(21+4i)$ $(1+i)(21-4i)$
$916$	$4+30i$ $30+4i$	$(1+i)^{2}(15-2i)$ $-i(1+i)^{2}(15+2i)$
$922$	$9+29i$ $29+9i$	$(1+i)(19+10i)$ $(1+i)(19-10i)$
$925$	$5+30i$ $14+27i$ $21+22i$ $22+21i$ $27+14i$ $30+5i$	$i(2+i)(2-i)(6-i)$ $(2+i)^{2}(6+i)$ $i(2-i)^{2}(6+i)$ $(2+i)^{2}(6-i)$ $i(2-i)^{2}(6-i)$ $(2+i)(2-i)(6+i)$
$928$	$12+28i$ $28+12i$	$-(1+i)^{5}(5+2i)$ $-(1+i)^{5}(5-2i)$
$929$	$20+23i$ $23+20i$	(p) (p)
$932$	$16+26i$ $26+16i$	$(1+i)^{2}(13-8i)$ $-i(1+i)^{2}(13+8i)$
$936$	$6+30i$ $30+6i$	$-i(1+i)^{3}\cdot 3\cdot (3+2i)$ $-i(1+i)^{3}\cdot 3\cdot (3-2i)$
$937$	$19+24i$ $24+19i$	(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)
$954$	$15+27i$ $27+15i$	$(1+i)\cdot 3\cdot (7+2i)$ $(1+i)\cdot 3\cdot (7-2i)$
$961$	$31$	(p)
$962$	$1+31i$ $11+29i$ $29+11i$ $31+i$	$(1+i)(3+2i)(6+i)$ $(1+i)(3+2i)(6-i)$ $(1+i)(3-2i)(6+i)$ $(1+i)(3-2i)(6-i)$
$964$	$8+30i$ $30+8i$	$(1+i)^{2}(15-4i)$ $-i(1+i)^{2}(15+4i)$
$965$	$2+31i$ $17+26i$ $26+17i$ $31+2i$	$i(2+i)(12-7i)$ $(2+i)(12+7i)$ $i(2-i)(12-7i)$ $(2-i)(12+7i)$
$968$	$22+22i$	$-i(1+i)^{3}\cdot 11$
$970$	$3+31i$ $21+23i$ $23+21i$ $31+3i$	$i(1+i)(2-i)(9-4i)$ $(1+i)(2+i)(9-4i)$ $(1+i)(2-i)(9+4i)$ $-i(1+i)(2+i)(9+4i)$
$976$	$20+24i$ $24+20i$	$-i(1+i)^{4}(6-5i)$ $-(1+i)^{4}(6+5i)$
$977$	$4+31i$ $31+4i$	(p) (p)
$980$	$14+28i$ $28+14i$	$(1+i)^{2}(2-i)\cdot 7$ $-i(1+i)^{2}(2+i)\cdot 7$
$981$	$9+30i$ $30+9i$	$i\cdot 3\cdot (10-3i)$ $3\cdot (10+3i)$
$985$	$12+29i$ $16+27i$ $27+16i$ $29+12i$	$i(2-i)(14+i)$ $i(2-i)(14-i)$ $(2+i)(14+i)$ $(2+i)(14-i)$
$986$	$5+31i$ $19+25i$ $25+19i$ $31+5i$	$(1+i)(4+i)(5+2i)$ $(1+i)(4-i)(5+2i)$ $(1+i)(4+i)(5-2i)$ $(1+i)(4-i)(5-2i)$
$997$	$6+31i$ $31+6i$	(p) (p)
$1000$	$10+30i$ $18+26i$ $26+18i$ $30+10i$	$-i(1+i)^{3}(2+i)^{2}(2-i)$ $(1+i)^{3}(2-i)^{3}$ $-(1+i)^{3}(2+i)^{3}$ $-i(1+i)^{3}(2+i)(2-i)^{2}$

Bibliografia

• GregG. Dresden GregG., Wayne M.W.M. Dymàček Wayne M.W.M., Finding Factors of Factor Rings over the Gaussian Integers, „American Mathematical Monthly”, 7, 112, 2005, s. 602–611, DOI: 10.2307/30037545, JSTOR: 30037545 .
• EllenE. Gethner EllenE., StanS. Wagon StanS., BrianB. Wick BrianB., A Stroll Through the Gaussian Primes, „American Mathematical Monthly”, 4, 105, 1998, s. 327–337, DOI: 10.2307/2589708, JSTOR: 2589708 .
• H.H. Matsui H.H., A bound for the least Gaussian prime omega with alpha < arg(omega) < beta, „Archiv der Mathematik”, 6, 74, 2000, s. 423–431, DOI: 10.1007/s000130050463 .

Linki zewnętrzne

• Liczby pierwsze Gaussa
• Rozkład na czynniki pierwsze