WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?
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1 WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they get mzng results. Here s toy s repe t Cfé Mth. Complex Integers 1 re efne s the set of omplex numers whose rel n mgnry prts re oth ntegers. For exmple, 4, 7, n 3 re omplex ntegers. A Rel Prme s n nteger > 1 tht s only vsle y 1 n tself. We n exten ths efnton to omplex ntegers n efne Complex Prme s omplex nteger whose moulus 3 s > 1 n s only vsle y 1,, n tself. For omplex prmes the sgns of the rel n mgnry prts n e postve or negtve. Complex prmes re symmetrl n the omplex plne. We only nee to test omplex ntegers where 0, 0, n n fn ll the others usng symmetry. Ths group of omplex prmes ± ±, ± ± re lle ssotes. Every omplex prme s vsle y some of ts ssotes; these o not ount s ftors. 1 These re tully lle Gussn Integers n honor of Crl Frerh Guss who ws the frst mthemtn to stuy them. Atully lle Gussn Prmes. 3 The moulus of omplex numer s n s wrtten. 009 Mrk Armrust Permsson grnte for eutonl use.
2 WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? Let s fn some omplex prmes. The smllest 4 omplex nteger > 1 s 1 n t s omplex prme sne there re no possle vsors etween t n 1. By symmetry, 1 s lso omplex prme. The next smllest omplex nteger s. (1 (1 so t s not omplex prme. ext omes. It s not vsle y ny of the omplex prmes foun so fr so t n ts onjugte,, re the next omplex prmes.? It s vsle y so t s not omplex prme. 3 s not vsle y ny of the smller omplex prmes so t s omplex prme. 3? It s vsle y 1. 3? Dve y the omplex prmes foun so fr: 1 ±, ±, n 3. It s omplex prme, s s ts onjugte, ? It s vsle y 3 n 1. Here re the frst few omplex prmes: (1 7 (10 9 (14 (17 (19 10 ( (7 11 (14 9 (17 8 ( (8 3 (11 4 (14 11 (17 10 (19 16 (3 (8 5 (11 6 (15 (17 1 (0 (4 (8 7 (1 7 (15 4 (18 5 (0 3 (5 (9 4 (13 (15 14 (18 7 (0 7 (5 4 (10 (13 8 (16 (18 17 (0 11 (6 (10 3 (13 10 ( (0 13 (6 5 (10 7 (13 1 (16 9 (19 6 (0 19 Rememer tht you n flp the sgns n swp the rel n mgnry prts to get other prmes. There s grph n the ppenx tht shows the lotons of omplex prmes n the omplex plne. Whh Rel Prmes re not Complex Prmes?, 3, 5, 7, 11, 13, 17, 19, 3, 9, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 There s pttern here. Rememer tht when you multply omplex numer,, y ts onjugte,, you get the rel numer. Ths mens tht ny rel prme tht s the sum of two squres n not omplex prme. 4 Smllest menng lowest norm. The norm of omplex nteger s. Ths s fferent thn the norm of generl omplex numer whh s the sme s the moulus:. 009 Mrk Armrust - - Permsson grnte for eutonl use.
3 WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? It turns out tht there s n esy wy to etermne f prme numer s the sum of two squres. Look t the struk out prmes n the ove lst n see f you n fn out wht they hve n ommon. (Ignore ; one gn t s the oll prme!) All of the o prmes tht re not omplex prmes re the sum of n even squre n o squre. Even squres ll hve the form ( n ) 4n for some nteger n. Ths mens tht they re ll multples of 4. O squres ll hve the form (n 1) 4n 4n 1. Ths mens tht they re ll multples of 4, plus 1. The sum of n even squre n n o squre must therefore e multple of 4, plus 1. We n rephrse ths so tht s lso exlue: rel prme s omplex prme f n only f t s multple of 4, plus 3. Is there n eser wy to tell f omplex nteger s omplex prme? We ve just foun smple test for omplex ntegers whose rel or mgnry prt s 0. Wht out the rest of the omplex ntegers? If omplex nteger z s prme then ts onjugte z s prme. Ths mens tht z n z re the only ftors of z z. If n re non-zero there re no rel ftors of z z. Therefore, f s not rel prme, z s not omplex prme. If omplex nteger z s not prme t hs ftors v n w n z z vv ww n w w re oth rel, z z hs two rel ftors n s not rel prme. Therefore, f rel prme, z s omplex prme.. Beuse v v s Hvng prove the test for oth ses, we n stte: omplex nteger z wth n non-zero s omplex prme f n only f s rel prme. Prme Ftorng Complex Integers When prme ftorng omplex ntegers we only wnt to use ftors n the form ± where n > 0 n <, n the ftor where n 1,, or 3. Usng ths stnr form fores unque ftorzton. Ftors tht re not n ths form n e onverte nto the orret form y multplyng y n. Here re some exmples: 3 4 ( ( (1 To fn the omplex prme ftors of postve nteger fn ts rel prme ftors. Chek f ny of the rel prme ftors re not omplex prmes. Reple them wth the omplex onjugte prs tht re ther ftors: 009 Mrk Armrust Permsson grnte for eutonl use.
4 WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? (1 (1 3 ( (1 3 (1 3 ( ( (1 3 ( ( ( ) ( (1 ) 3 1 (1 4 3 Ftorng negtve nteger s the sme exept tht t gets n ftor: 5 ( ( To ftor pure mgnry nteger, ftor s f t were rel nteger n multply y : 5 ( ( Ftorng omplex numer s t tougher. We nee few prelmnry onepts to e le to o ths. The norm of omplex nteger s efne s (. (z) s multpltve, tht s ( w z) ( w) ( z). 5 Gven omposte omplex nteger z p p p z p p 1 n, ( ) ( 1 ) ( ) ( p n ). ote tht the ftorzton of the norm gves the norms of the ftors. Determnng the ftors s not qute utomt sne there re multple omplex ntegers wth the sme norm. Also, ny prme nteger q tht s ftor hs q s ts norm. Ftor z ( z) Ths ftors to Ftors wth even powers re lkely the norm of pure rel ftor. ( 3) 3 so 3 s lkely ftor (ovously, n ths exmple). z Try to prove ths entty yourself. A proof s shown n the ppenx. 009 Mrk Armrust Permsson grnte for eutonl use.
5 WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? onfrmng tht 3 s ftor. Complex prmes n stnr form whose norms equl 5 re ± oes not work, ut We n onvert the fnl ftor nto stnr form y vng t y Ths gves the ftorzton ( ( Ftor z ( z) 8450 Ths ftors to The frst ftor hs norm so t must e ntes tht 5 s lkely ftor, ut 5 s not omplex prme so we nee two ftors of the form ± , 19 oes not work, so n re ftors. 13 s not omplex prme so we nee two ftors of the form 3± so there re two 5 Ths gves the ftorzton (3 3 ftors n fnl ftor of. ( 1 ( ( ( 3 ) Mrk Armrust Permsson grnte for eutonl use.
6 WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? 009 Mrk Armrust Permsson grnte for eutonl use. A p p e n x Complex prmes wth moulus less thn 100 on the omplex plne: Proof tht (z) s multpltve: ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) z w w z
7 TI-83/84 Progrm to Compute Gussn (Complex) Prme Ftors 1 Prompt Z Z0 Z 3 6 L 4 (s(z)) M 5 1 F 6 Ll 7 0 E 8 Z/F W 9 Whle fprt(rel(w))0 n fprt(mg(w))0 10 E1 E 11 W Z 1 Z/F W 13 En 14 If E 0: Then 15 Dsp {F,E} 16 L-1 L: If L 0: Puse 17 (s(z)) M 18 En 19 If mg(f)>0: Then 0 onj(f) F 1 Goto En 3 onj(f) F 4 If mg(f)<rel(f) n s(f) M 5 Goto 6 rel(f)1 F 7 If fprt(f/)0 8 F F 9 If s(f) M 30 Goto 31 0 E 3 Whle rel(z)<0 or s(rel(z))<s(mg(z)) 33 Z/ Z 34 E1 E 35 En 36 If Z 10: Then 37 Dsp {Z,1} 38 L-1 L: If L 0: Puse 39 En 40 If E>0 41 Dsp {,E} otes: Lne onverts Z nto omplex numer f rel numer ws entere. Ths s requre to prevent Dt Type errors n omprsons. Ths s the sme reson lne 36 ompres gnst 10. Lnes 6 through 30 re the ftor test loop. The ftors teste re 1±, ±, 3, 3±, 4±, 4±3, 5, 5±, 5±4, In prtulr, there s no nee to test ny Gussn nteger whose omponents re oth even or oth o. (Just s s the only even rel prme, 1 s the only even Gussn prme.) Throughout the ftor test loop, M s the squre root of the moulus of Z. o ftor of Z n hve moulus greter thn M. The test gnst M n lne 4 elmntes out 1% of the trl vsons, s shown n the fgure to the rght.. When the ftor test loop exts, Z s prme or power of. However, Z my not e n the form we wnt; ts rel prt my e negtve or ts mgnry prt my e greter thn ts rel prt. Lnes rotte Z untl t s n the orret form, keepng trk of the power of neee s the fnl ftor.
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