48 wsa so 5 2014 Dgl/ «?Hf« Ve mot so''''day so GO chifa cac so''''nguyen to ThS LE QUANG HUY LU(JNG TU HANH Trudng Dpi hpc Hdng BQc I BAT VAN DE Trong chuong trinh Toan phd thong, HS da dupc lam quen vdi c[.]
48 so 5-2014 Dgl/.«?Hf« wsa Ve mot so'day so GO chifa cac so'nguyen to ThS LE QUANG HUY - LU(JNG TU HANH Trudng Dpi hpc Hdng BQc I BAT VAN DE Trong chuong trinh Toan phd thong, HS da dupc lam quen vdi cdc kh^i niem "day sd" nhu: Day s6 ty nhi6n each dfiu, ckp s6 cdng, cSp s6 nhSn vd mdt sd day sd khac bac DH v^ CD, cdc ki€n thiic vk day sd duac nghien cuu mpt each cd hS thong, giiip SV hilu sau kien thuc da hoc, Khdi ni?m "s6 nguyen td" da duoc dua vao nghiSn ciju va ling dung chucmg trinh Toan THCS, THPT cung nhu d DH v^ CD Tuy nhien, thuc te cac van dk nSu trSn dugc nghiSn cUu mdt each rieng biSt Bai vi^t dudi ddy cdc tdc gia da "l6ng gh^p" khai niem dd vdi bang cdch dua m§t s6 tinh chat ciia day sd cd chiia sd nguyen td II MOT S TINH CHAT CO BAN CUA DAY SO Chiing ta chi x^t c^c ddy sd md tdt ca sd hang cua nd deu la sd tu nhien, Moi d5y sd nhu vpy la mdt anh xa M : N ^ N, dupc xdc djnh bdi n H> w(«) = H„ Ky hieu M„ la sd hang t6ng qudt cua day, Menh tte 1: Khdng tdn tai m^t cdp s6 cdng cd cong sai khac 0, cd vd sd s6 hang vd mdi s6 hpng d€u Id cdc sd nguyen td Chuvg minh: Gia sir tdn tpi mdt cdp sd cdng u^,U2, ,u^ \a\ cdng sairf?f cd tat ca cdc s6 hpng la sd nguyen td Theo cdng thiic cua s6 hang tdng quat cdp sd c^ng, ta cd t/^ = Wj + (« - i)d Vi W^ E N vdi moi n ty nhien nen a ' > l , K h i d d u^_^^=u^+iu^+l-l)d^u^(\ + d) chia 6r +1,6/ + hodc 6r + vd tich ciia cdc sd tu nhien dpng (>t +1 ho3c dr + khdng cd dang 6m + nen cdc udc cua a phdi cd udc dpng 6/ + 5, Gid sii d = 6t-v5 \t mdt udc ciia a, d >l Nlu d Id sd nguyen td thi ta cd dpcm NIU ^ Id hpp sd thi lap luan nhu tren ta dupc d, = 6(| +5, (| £ A', rf| > 1, Id udc cua d va ciing la udc ciia a d8u Id sd le, suy mpi udc Ciia a Aku cd dang Khi dd nlu a"" = (mod;?), m e N* thi m\d L^p lu^n tuong ty ta duac tf, la sd nguySn td ho^c £?, cd udcrfj= 6/3 + 5, (^ e N Vi o >rf,>rf^> ndn qud trinh tren phdi kit thiic sau mdt sd hiiu han budc vd ta dupc dpcm Trd lpi chiing minh menh de Gid su chi cd n s6 nguyfin td dang 6m + 5, m e N la Pi,Pi,-,Pn , d^t fl = ^PxP2-.p„ - i - Ta cd a = ()p^p2 p„-l = (>{p•^p•2^ p„-\) + 5, suy a cd dang dm + 5, m e N Theo chiing minh tren thi fl cd udc nguyen td p = 6A: + 5, vdi k la mdt so ty nhien ndo dd Vi chi cd n so nguyfin td dang 6m + 5, m e N, la Pi,p2, ,P„, n^T^ i^n Xai i,\Mj, d>l, nen u^^^ sai d = chiia vd sd sd nguyen to la hpp sd, vd ly, suy bd de dupc chijng minh, Chiing minh tuong tu ta cd menh dl sau Tuy nhien cd nhirng cap sd cdng chiia vd sd sd Mf/i/i 3: Trong cdp s6 cdng «, = , u^,.,,, u„, , nguyen td, ta cd m^nh 6k sau cdng sai cf = cd vd so so hang la cdc sd nguySn tfl M^nh de 2: Cap sd cdng vdi s6 hang ddu ti6n bdng Mpdi de 4: Cho p la m p t ^ nguyen td, ;? > , Tdn tai ddy va cdng sai bang chiia vd sd sd hang la sd nguygn td Chung minlr Trudc hit ta chiing minh mdi sd tir vo hpn cdc sd nguyen td phan biet ^, p^^ , /j„, thod man nhiSn a = 6m + 5,me N, d£u cd it nhat mdt udc nguyen diluki?n p^ = (modp"), vdi mpi « = I,2, td ;j = d« + 5, vdi n la mpt sd ty nhien nao dd Chdiig minh: Trudc hit ta chiing minh bd dl sau: Vi m e N nen a>\ Bo di 1: Cho a Id mpt sd r^uyen, ;j Id sd ngi^n td ciing N6u a Id sd nguySn td thi ta cd dpcm vdi a vd (/ Id sd nguyen duong nhd nhat ch.o t/=\ (n»d p) Gid sii a Id hpp sd Vi a Id sd le nSn mpi udc ciia a DiflivdHOC 505-2014 That vay, Idy a chia cho d, gia sii a=dq-¥r, vdi 0 bd da diing vdi n=k, m c6 NIU k X^t so hang P/^ ^, Tir gid ta thilt suy p^_^j-2p^=l ho$c pi.^.,-2pj=-l, Gid sd p^ Id udc nguySn td nhd nhat ciia 5„_j Vi Vi pj^ Id so nguyen td, p, > 3, nan pj =1 (mod 3) S„_, Id udc ciia A^ nen p„ cung Id udc nguyen td ciia A^ , hodc pj s - (mod 3), ta chiing minh /4„_, khdng chia het cho/>„, Nlu p,sl(mod3)thipi,,-2p^ = - l pj,,=2p,-l Gia sii ngupc lai A^_^ chia hit cho p„ Ddt Khi dd ta cd a=l(modp^) Suy o'=l(mod/7„) , vdi mpi i e N Ta cd B,,_^=2_^a'=p(modpJ That v§y, neu p^^j - 2pj = thi pj_^, = 2pj^ + , Tii p^ =1 (mod3) suy p^.^., = 2p^ + chia hit cho 3, vd I^, a = 2^ ta dupc A^_i ~ a-l Vi B^_i-p„ nen \i Pi^_^j la so nguyen td Idn hon Tuong ty, neu p^s-l(mod3) thi p^^j=2pj + l Ta xdt cdc trudng hpp sau: Trudng hap i: p, sl(mod3), dd theo chung p:p„ E)o p, p„ la cdc sd nguyen td nan p=p„ => minh tren thi p^ =2pj -l=l(mod3), Bdng quy nap ta a=l(modp)(3), chiing minh dupc p,, =1 (mod 3), vdi mpi k G N', suy Vl /? Id mgt sd nguyan iQ, p>2 nen (p, a)=l Ap dung b d l t a c d 2^' =2 (modp), suy os2(modp)mdu thuSn vdi (3) V|y A^_f khdng chia hit chop„ Tii cdch xay dung ddy A^, A^-^-, ^„,—ta thdy nlu n>m thi A^\A^ vd A^'-P„ nhung A^ khdng chia het Pi^^^ = 2p( - , vdi mpi k e N* Ta cd pt = 2*="' pj - (2*"' -1), VA > I (6) Thpt vdy (6) diing vdi k-2 Gid sd (6) ddng vdi k=m, nghia Id ta cd p^ = " p, - (2 -1) Ta cd : p , =2P -1= 2(2""'p, -(2""' -1))-1=2";>, - ( " -1), cho p„ Suy day Pj, P2,—> p„,— la ddy vd hpn cdc suy (6) cung d£ing v6i k=m + l V^y (6) dung vdi m^i sd nguydn td phdn bift Jt>l 50 W^%s:^WfWM Tir (6) suy P\>'i nan p4=-(2 thay -l)(modp,) A - p , vdo (7) wm (7) Do ta duoc SO 5-2014 7q^.j VmvaHoc Vi 7^ = (mod4) nan 7*' = (mod 4) Khidd5.3.2.p4,.,pk-i i 4hay5,3.p4 pk-i '• ( ) Vi Nk = 5.3,2.p4 pk.i + Id sd le nen pk Id sd \h \is\ p ^ s - ( ^ " ' - l ) ( m o d p , ) Theo djnh 1^ Fermat ta cd mpi k > , suy (1) khdng xdy ra, Vdy p^^^Tj^nst^ 2'^"' s i ( m o d p , ) , suyra p ta cd dpcm Gid sii tdn tai k, k > cho pk= 11 Vi Nk = p[P2 pk-i + = 5.3.2.p4 pk-i + khdng chia hit cho 5, 3, 2, vd 11 la thira sd nguySn td Idn nhSt ciia NknenNk=ll^ Khidd5.3,2.p4 pt-, = l l " - 1, suy 11"" s (mod 3) => k = 2t, v d i t G N" Vi 11^ = (mod4) nan 1 ^ ' = (mod 4) chia het cho p , , vd 1;?, Trudng hap 2: P | = - l ( m o d ) Chiing minh hodn todn tuong ty ta dupc p^^j =2p4 +1 vdi mgi A: e N TU Pk = pj + (2 " - ) , VA > ta cung suy dupc p chia hit cho p , , vd 1;^ Vdy menh de dupc chiing minh, M^i/j d i (i: Ddy sd cdc sd n g i ^ n td p^, pj_, , p,^, A\sgc xdc dinh nhu sau: p , = , p„ Id thUa sd nguyen td Idn nhat ciia sd l + p,p2 ,p„_[, vdi n>2 p„ ^11, Vn e Khi ta cd N' Chung minh: Ddt N^ = ptp2, pj;.i + 1, vdi mgi k > Khi dd pk Id thira so nguyen td Idn nhat cila Ni(, vdi mgi k>2 Tacdpi = , N - + l - => P2 = 3,N3 = 5,3 + 1=16 =e> p3 - 2, Trudc hit ta chimg minh p^ ^1, Vn € N*, Gid su tdn tai k, kS:4 chopk = ViNk = piP2 pk-i + =5.3.2.p4 pk.] + khdng chia hit cho 5, 3, vd Id thira sd nguyan td Idn nhdt cua N^ nen Ni, = 7*' Khidd5.3.2,p4 pk.i = ' ' - i , suyra?*^ = (mod 5) => k = 4t, v d i l e N* > fix I Khidd5.3.2.p4 pk-i : hay 5.3.p4,,,pk.i : 2(2), Vi Nk = 5.3.2.p4 ,pk.i + Id sd le nan pk Id sd le vdi moi k > 4, suy (2) khdng xdy Vdy p^ ^11, VM e N vd ta cd dpcm III K E T L U ^ N Khi hpc vl sd nguyan td, HS bilt den djnh 1;^: Cd v8 sd sd nguyen to Djnh ly ndy de cap den mdi lien h^ gitia khai niSm "sd nguySn td", "ddy sd" vd bdi vilt dd gidi thieu tham mdt so tinh chat cd npi dung trSn Hy vpng dd se la tai lifu bd ich cho HS, GV vd nhiing quan tam din vdn da ndy nghien ciiu, tham khao TAX LIEU T H A M K H A O 1, Lai Dtic Thjnh S6 hpc NXB Gido diic, H 1997 2, Nguyen Tiln Tdi (chii bien) - Nguyin HQu Hoan So hpc NXB Gido due, H 2000 3, Bdi Huy Hiln - Nguyen Hij-u Hoan, Bdi tap Dai s6 va S3 hpc (tdp 1) NXB Gido due, H 1985 4, Nguyin Tiln Quang Bdi tap s6 hpc NXB Gido due, H 2001 S^' i' W va THi; Mdl VIET BAI, TIN, ANH D A N G TREN TAP c m DAY vA HOC NGAY NAY Tap chi Day va Hpc xuit ban mdi thdng mdt sd, 76 trang, khd 20,5 x 28,5, rudt in mdu, bia mdu, gia 20.000d/cudn, Tap chi Iudn Iudn phan dnh nhihig vdn d^ mdi, hien dai cua II Iudn vd thUc tiSn vi phuong phap day vd hpc, ve tu quan li giao due, v6 dudng va hinh thtic jAiat triln cong cudc khuydn hoc, khuyen tai vk xdy dung xa hdi hoc tap cf Vi6t Nam T a p chi Day va Hpc trdn kfnh mdi cdc thiy cd gido cdc tru&ng Tidu hoc, THCS, THPT, THCN, CD, DH, cdc Trung tdm GDTX, Hdi Khuydn hpc cdc cdp„„ gtti bai, tin, dnh dang tai trdn Tap chi, dd cdc thdy cO giao trao ddi kmh nghiem vdi ddng nghiSp qua trinh giang day, nghien cihi, hoc tdp Bdi, tin, anh xin giii theo dia chi: T A P C H I DAY VA H O C N G A Y NAY Nha sd 1, ngo 29, phd Vong Thi, qudn T a y Hd, H a Noi Didn thoai/fax: 04.37531012 - DD: 0983081976 * Email: dvh_nn@yahoo.com Xin trdn trpng cam on! 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