ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) Floet Smdche, Ph D Aocte Pofeo Ch of Deptmet of Mth & Scece Uvety of New Mexco 2 College Rod Gllup, NM 873, USA E-ml: md@um.edu I th ecto peeted ew tege umbe lgothm fo le equto. Th lgothm moe pd th W. Sepk peeted [] the ee tht t eche the geel oluto fte mlle umbe of teto. It coecte wll be thooughly demotted. INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS Let u code the equto (); (the ce,b,, educed to the ce () by educg to the me deomto d elmtg the deomto). Let d (,..., ). If d b the the equto doe ot hve tege oluto, whle f d b the equto h tege oluto (ccodg to well-kow theoem fom the umbe theoy). If the equto h oluto d d we dvde the equto by d. The d (we do ot mke y etcto f we code the mxml co-dvo potve). Alo, () If ll the equto tvl; t h the geel tege oluto k,,, whe b (the oly ce whe the geel oluto -tme udetemed) d doe ot hve oluto wheb. (b) If, uch tht the the geel tege oluto : k b d x k,,..., \ The poof of th eto w gve [4]. All thee ce e tvl, theefoe we wll leve them de. The followg lgothm c be wtte: Iput A le equto: (2) b,,b,,,, wth ot ll d (,..., ).
Output The geel oluto of the equto Method. h :, p : 2. Clculte m, (mod ), d deteme d the p, (, ) fo whch th mmum c be obted (whe thee e moe poblte we hve to chooe oe of them). 3. If go to tep 4. If, the x : t x b h x : th x b (A) Subttute the vlue thu detemed of thee ukow ll the ttemet (p), p,2,... (f poble). (B) Fom the lt elto ( p) obted the lgothm ubttute ll elto: p, p 2,...,() (C) Evey ttemet, ttg ode fom p me pocedue (B): the p hould be ppled the 2,...,(3) epectvely. (D) Wte the vlue of the ukow,,, fom the tl equto (wtg the coepodg tege umbe pmete fom the ght tem of thee ukow wth k,...,k ), STOP. 4. Wte the ttemet (p) : x t h 5. Ag x : t h h : h : p : p The othe coeffcet d vble em uchged go bck to tep 2. The Coecte of the Algothm Let u code le equto (2). Ude thee codto, the followg popete ext: 2
Lemm. The et M, (mod ), < h mmum. Obvouly M * d M fte becue the equto h fte umbe of coeffcet:, d codeg ll the poble combto of thee, by two, thee the mxmum AR 2 (ged wth epetto) = 2 elemet. Let u how, by educto d budum, tht M Ø. M Ø (mod ),,. Hece (mod ),,. O th poble oly whe,,,, whch equvlet to (,.., ),,. But (,.., ) e etcto fom the umpto. It follow tht,,, fct whch cotdct the othe etcto of the umpto. M d fte, t follow tht M h mmum. The Lemm 2. If m M, the,,., We ume coveely, tht, uch tht.,. Let p, p, uch tht p d m p ot dvded by. Thee coeffcet the equto, whch the mmum d the coeffcet e ot equl mog themelve (coveely, t would me tht (,.., ) whch gt the hypothe d, g, of the coeffcet whoe bolute vlue gete tht ot ll c be dvded by (coveely, t would mlly eult (,.., ). We wte p / q (tege poto), d p q. We hve p (mod ) d. Thu, we hve foud whch cotdct the defto of mmum gve to. Thu,,. Lemm 3. If m M fo the p of dce (, ), the: 3
x t k b h x th k b x k Z,,..., \, the geel tege oluto of equto (2). Let x e x e, e,, be ptcul tege oluto of equto (2). The k x tht: d,,..., \, d t h x x x x x b x x x x x b x x k x,,..., \,. (becue M ) uch Lemm 4. Let d, be the p of dce fo whch th mmum c be obted. Ag, let code the ytem of le equto: (3) t x x b h t x x h The x e x e, e, ptcul tege oluto fo (2) f d oly f x e x e, e,..., \ d t h t h the ptcul tege oluto of (3). x e x e, e, ptcul oluto fo (2) f d oly f 4
x b x x x x b e e e h t x x b d t h x e,..., \ d t h t h ptcul tege oluto fo (3). x e x e, Lemm 5. The pevou lgothm fte. Whe the lgothm top t tep 3. We wll dcu the ce whe. Accodg to the defto of, *. We wll how tht the ow of uccevely obted by followg the lgothm evel tme deceg wth cycle, d ech cycle ot equl to the pevou, by. Let be the ft obted by followg the lgothm oe tme. the go to tep 4, d the tep 5. Accodg to lemm 2,,,. Now we hll follow the lgothm ecod tme, but th tme fo equto whch (ccodg to tep 5) ubttuted by. Ag, ccodg to lemm 2, the ew wtte 2 wll hve the popety: 2. We wll get to becue d, d f, followg the lgothm oce g we get epetto. d o o. Hece, the lgothm h fte umbe of Theoem of Coecte. The pevou lgothm clculte the geel oluto of the le equto coectly (2). Accodg to lemm 5 the lgothm fte. Fom lemm t follow tht the et M h mmum, hece tep 2 of the lgothm h meg. Whe t w how lemm 3 tht tep 3 of the lgothm clculte the geel tege oluto of the epectve equto coectly the equto tht ppe t tep 3). I lemm 4 t how tht f the ubttuto tep 4 d 5 toduced the tl equto, the geel tege oluto em uchged. Tht, we p fom the tl equto to le ytem hvg the me geel oluto the tl equto. The vble h coute of the ewly toduced vble, whch e ued to uccevely decompoe the ytem ytem of two le equto. The vble p coute of the ubttuto of vble (the elto, t gve momet betwee cet vble). Whe the tl equto w decompoed to, we hd to poceed the evee wy,.e. to compoe t geel tege oluto. Th evee wy dected by the ub-tep 3(A), 3(B) d 3(C). The ub-tep 3(D) h oly ethetc ole,.e., to hve the geel oluto ude the fom: f (k,...,k ), 5
,, f beg le fucto wth tege umbe of coeffcet. Th f poble how tht ubttuto e ot lwy poble. But whe they e we mut mke ll poble ubttuto. Note. The pevou lgothm c be ely toduced to compute pogm. Note 2. The pevou lgothm moe pd th tht of W. Sepk [],.e., the geel tege oluto eched fte mlle umbe of teto (o, t let, the me) fo y le equto (2). I the ft plce, both method m t obtg the coeffcet fo t let oe ukow vble. Whle Sepk tted oly by chce, decompog the getet coeffcet the module (wtg t ude the fom of um betwee multple of the followg mlle coeffcet ( the module) d the et), ou lgothm th decompoto ot ccdetl but lwy eek the mllet d lo chooe the coeffcet d fo whch th mmum cheved. Tht, we tet fom the begg the hotet wy to the geel tege oluto. Sepk doe ot ttempt to fd the hotet wy; he kow tht h method wll tke hm to the geel tege oluto of the equto d ot teeted how log t wll tke. Howeve, whe lgothm toduced to compute pogm t pefeble tht the poce tme hould be hot poble. Exmple. Let u olve 3 the equto 7x 7y z 2. We pply the fome lgothm.. h, p 2. 3, 3, 2 3. 3 go o to tep 4. 3 4. () y t 7 z t z 5. Ag y : t h : 2 3 : 3 p: 2 wth the othe coeffcet d vble emg uchged, go bck to tep 2. 2.,, 3 3. x ( 3t 2 ( 7t ) 2) 3t 2 7t 2 z 7t 2 ( 7t ) 7 ( ) 3 7 ( 2) 7t 2 42t 72 3 6
(A) We ubttute the vlue of x d z thu detemed to the oly ttemet (p) we hve: () y t z 7t2 43t 72 (B) The ubttuto ot poble. (C) The ubttuto ot poble. (D) The geel tege oluto of the equto : x 3k 7k 2 2 y 7k 43k 2 72 z 7k 42k 2 72; k,k 2 REFERENCES: [] Sepk, W, - Ce ştm ş ce u ştm depe umeele pme? - Edtu Stţfcă, Buchet, 966. [2] Cegă, I., Czcu, C., Mhuţ, P., Opţ, Gh., Co Rehe Itoducee î teo umeelo, Ed. Dd. ş Ped., Buchet, 965. [3] Popovc, C. P. Atmetc ş teo umeelo, Ed. Dd. ş Ped., Buchet, 963. [4] Smdche, Floet U lgotm de ezolve î umee îteg ecuţlo le. 7