iNry
H0r ,r0AN tt()c
VIfT NAM
BAC RA
Chtli tthiQnt. NGUY&N C,iXU T0An-
I'ru s&: 70 'l'rdn liung Dao * Ha s*di
"i+rrtur#liflii[Hiir:iliitliqf
HAt IHANG MQT K'i
't'ha kg {de *oqn; E0AnG C}tfiNG
llav u0i i s262$
i:i!::i!:!i
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I
I\6i ehrryQn viri oAc biln tr6 yeu ro6rr
do u,
u(u - 1)
tdt. cd cdi s6 cd clqtg r.u - ---z , trong
u ld htri sd lV nhi6n thda ntttn iti|u l:i€n :
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s6" NGUYEN 16 MEC.XEN.NC}
ED.io rHr"l'T. sd l,'1981 ccr bai thong h'io vdi
lD cic lrrr ir r e s6 ttcttten t6 trrn ntrdt duoc
bidt. d.r I *', 22tz0t - l, do Nikel vi Noll
tim ra n[m 1979. Nhidu .ban 1'6u toin c6 r'i6t
thu h6i t6i vE j'nghia cira k6t rJuii ni1-'
C'.ie ban ili bi6t vai trd quan trcng ofrn ciitt
s6 nguyda td' Ilrn quan treng cang t6 rO hon
r,oi sn phit tri6n cria If thul'dt s6 v& sg xAm
nhip cfra I,v lhuydt s6 vAo c6c ngi.nh to6n hoc
khAc. Vi viv tr)' xua i16n n-ay, circ nh{ toAn hoc,
1r1' th€ h6 ni1, sang th€ hQ khnc, de b6 nhi€u
cdng sric r1b' nghi€n cd"u r--Ac s6 ngul'tin t6.
CAeh diy hrvn hai nghirr ninr, nhA todn hoc
cb dai O-clit d{ chung nrinh rr'rng c6 vd sd
s6 ngu1.6t t6. CAc Lran dlir hi6t chung minh cfia
0-cli!, d6 lir mOt cAch chr?ng minh kiFtr mltr
v0 st tbn ttri cira mdt d6i tuong toin hoc, tuv
hh6ng cho ph6p chi ra cu thB d6i tuong {y.
V{n dE rliu titn dugc tIXt ra iA tirn clch
< toin gon lt 1{t'cfl r:6c s6 nguy{p t6. 0 dAy c6
PHAN O[t( CUilVt-l
kdt qui nho r;h6 : dr)ng phtrcrng phip ( sArrg ) !
CA.c ban dA bidt ki6u < sdng E-ra"t6-xten >: vidt
c'irc sd tg nhi€rr tu 2 tr6 di thinh dfl-v, sd ? 1A
nguv6n t6; ta sing ltot thu nhdt : gqch t1i c6c
l6i -efra 2. sd dhu tien con lai (sd $) lt nguy€n
t6; ta sAng luot thri hai: gach di eAc bQi crla
3, s6 ilhu ti6n cdn tai (srl 5) lir nguy0n t6; ta
siug luot thri ba: gach dr cac bbi cta 5, s6 ilhu
ti6n cdn lai (e6 7) lA ngu-r-0n t6, vi cfr th6 m5i.,.
Xirr gi6i thi(u vOi cac bAn m6t ki6u silg
kh6c; coi nhu bii tAp:
RAI TOAN. T'roug tldg ltt nhian, ta b6 di:
-' IAt cd ctic lfiir th*a c&.a 2 :
.,o o1 ,,9,13 ,n
r t< I .
\
Chl"n..t! minh rdnq sou Alj trtoi lri:r (nc r<l dri
tftLt, !rong clntl sd iry nh,ir ri t trri lti chi tirt
lai td.t'e& cac sd nglugdrt td li"
Nhu cAc.batr thir', r'ac kidu sl'rng nd1, hrri ..
thri cdng nghi0p !!l D[ng hcrn, chtng kheng ehu
phep phit hi6n tti€rn rn6t tinh -Eh{1 i;do ctd cri:r
sd ngu-vCn t6 eo thb tlong gop cho srr phrit lr;6n
cria Iy thuydt sd ciing nhu c*a toan hoc.
Crar: nhi torlin boc thdi trudg rti titng co f
nghi: kv hiOu pn lir sd ngi:ye,i td lhLi n. tu'c lA:
fl : 2, pz* ?t, ru:5, p4=7, pi - 11, p6 =- 13....
hiy tim rnQt cdng th:.;re do'n gidri bibu diin pn
qua. n.,rchua cln. n6i ri thd ndo ld cdttr; thtrc
(r,i n6 con phu thudc ','ao tii;;h do lrtrii tri6rr.
eria toAn hr.rc), lhung chi r,6i ii€u chuhn don
gidn, vcti t6[ cii moi nii*n rihrcrng e6 thE *hdp
nhin iluoc, thi 1 nghi iray r;Qu ciu n6i trerr di
th{t bai : cho i[6rr nav nguoi ta ( c|ira n tim ru
diroc,.vd hi€rr rtav nguoi ta dn b6 i dinlr ( di€n
16 )) 11i tirl nrdt c0ng thut: d<rn giAn nirrr v$1'l
DArrh ha th{p r-0u rlau: lim rnrrl- c6n€ llruc
rloii' gi&n, phu-thu0c vi9 s6 tr.t trhirrn tr" rho ta
nhiing sd ngu3,€n t6 r,(ri rnoi n = l, 2, :'),.,. lihi
d6 ta sE dunb m$t df;v vd han, gdrn ioir iilii'ng
s6 ngul'6n i6. Tuy 6 tlit;' ta iiliO:rg iirrgc. tdl cri
c6rc ri6 ngur,€;: t0, rhnng m6t cdng iLiile d,,r r:
gi{n iy (n6u c6 l) cirac clrin s0 dong gop niridLr
eho 1y thuydt s6 cdng nhi6u ngA,irh tcAr-r l:oc
hh6c.'fi€c tha-v diiiu,do d6.n nay vSri chria thdrrh
su thrl.:
Cf.ch dA,t' l-ra triirn nlnr, nhA toAn hor Fec-ma
di nghi rir:g dtry si (s;;u nAy foi lri rJri,r' irec-rna)
prn = 2?nr i (ri ,= $, l, 2,. "l
giu'i toitn nhtr$g s6 ngu-r'0rr t6: irec-r-na clii tliu
vAi s6 diiu Fo = J, ,Fl : 5. Fzi 1;, i'3 :257,
Fa : 6.;5i17. Nhung n[rn i732, nhd toan hoc O-le
phnt hidn ra ri ng s6' .F5:4 294 967 297 la in,,t
tigp s6: quir v.i-v. p5: fi4 1.6700117.
Sd ngu1,6n id !'ee-ma iluo-i: d[e bigt chu 1-
n$tr mdt kdt qui'tuv6t rlrQrr uria nha toirn hoc
Gau-xo, gi&i guydt m6t b*ri loAn ddt ra tt thoi
cb ttai'-db chia rlao'c d.ttbng trbn bdng thuse ua
{rompo ,'e n phdn bdn|] nhatt, di-bfi k?rl cdn uci
d& la trong phdn tich sd n ru fhrru sd ngugdnN4,
kh6ng c6 ctic ihia sd n'gug|n tii.khac udi 2 ud uoi
' cric sd ncluy{n td !-ec-ma.
Khdng. c6i m6t phro'ng phirp dcrn gi&n dE bidt
flrrqc trong eric s6 Fer-rna. sd aAo li ugr-r1l6lr
tij, s6 niro lii hop srI. Or) eal i:<1 rn0l kdt r;ui
il{ng'luu i:
;.hhtr rlA),tninir di l: clio ia m&t ili6r.r ki€rr cdn
dE s(i lin li neLrr.gir tn. tii,,ri S,irt lilrong phrii
tA mdt diFr.r ki6ri dri. N'li cach khdc. rn€nh dE
1 cho ph6p.ph5t hi{n inOt s<j rhirrrg sd Fec-rna
hop s6, ndrr th,.rc hi0n rlur.r't ph0p chia hai sd
noi trin vd ph6p chia l:1 cd drr. Er,'rng l& viOe
phAt tiiQn cltc's6 lier-:rru lr,.rtr so c6 ph.hn ( cii
dang > ho n vi6c phiit Ir i0 n lic si Fec-rna ngriydn
td: cho d€n nav, ngud'i la iih ng bi6t cdc s6
Fec-tna r;gu-r.6rr t6 nio. ngoii 5 -.ii <r trim thrrrrng ))
Fn r'6i !4 = 0, 1' 2. fl. 1, tlonc Lrjc rio rrqud'i ta
bi6t ghn'10:;6 Fec-rna hr-rp' ;r':. chring han ciic
s6 -Fn r.6i 5 ( ir .s 16 ddu liL lhn,rg hop s6. 56
Ft7 cb 3911r? chii sd (trnng dang thqp phin),
lii0ti n:rv {riic raay iilh di0l, iLi. citrra cr'r kh&
or" '"
"r3i o?1
i,iri! larl ph4p r:hi;r so 3' * i = ,t" + 1
(c6 i:<in r339 4ro eht sd) chr; so Fg7 dF sii. clung
r.lrro-c mQnh rlB i, I i vay cLo ddl ria;r riguor ta
chua bi6l I't: ld sd ngu r.0n td har. l,i hop s6 :
nd.r ching hai: lla,,'ir,5ri la. Frr lA nqur6:t. 16,
thi n6 s0 iA sd nru-r'6n 1d iorr iron rrit uhiEu
. ..2 1?01
'so vr;i so l--'"' - 1. sd nr)r. chi c6... 6o33 chiL
s6 I Sd litgas (c0 hon 10584 "chi, sd) lA hup s6:
c1i: thui:g to di&u d6, ngueri ia khdng ttr6 sri.
u,,irrg i1ig1,11 ,cl$_ i. ma chtil-.g nrinh ring n6 chla
ntt ch., 5'l'"= t 1 iso r:d.. li ngur.in i6). Ili,v
cirrg li piruulg phlp chung clE chung tir mirt
ljn nAo Co. la hop so: nhu ra.r;' tir hi6u- rlog
vi6c Ap duirg mdnh riF 1 rdt han ch6.
iJirl iree-nru li mdt dir- cori t.iia di-r'
un = 2L * i (p : l. ?, 3,...i (1)
Ndlt :r eirirn nrdt ihiia sd 16 lor: irryl i. ching
irar.r n: kcl. voi r/ itr l6n ho' 1, ta rhLrng rnirrh
dc dnng ring ur: rkd + 1 chia h6t cho 2k * l,
lhrr vAv n6 ld mQt hqp s6. N6i cAeh khic,
da]. (J) chria vd sd nhrlng hqp si,. cic sd nguy€n
t6 ircrrg de1, (1) chi cr1 thft la nhtna s5 u1 i'r'g
vcvi n khOng eri thua sd 16, lr?c li:r n = 2"', khi
tlo iin :22*+ i = Fr*. 'l-u,rr lal trrrlg ctlL-r ( j )
. clio ddn narl' ngiroi irr rrtrii chi. bi6t v6l ',.en eo
! sii rrgur 6n lri I
. Dey (t) goi -i day
. &!yr= 2n : 1(,i : i.:" 1J,.. i, e,,i 11 d1g Mec-a'en-
na. lthrr vir'
lrll.= I. i/ j = J, "1,'3 = 7. i/a - [5, .1/5 = 31.
Mc = 63,...
N€lu n la nr6i hop sd, chnng hqn n H aD, voi
c, b iA airiing s6 ttt nhiOr: ldr hon 1 vA r;h6
ho'rr n, thi IS; : 2'I - I ehia hCt cbo 2a - l, d.o
dd Mn Id mdt ir<rp s6. L&p ludn nAy e hriag ti.r
rii*g dly Nlcc-xen-no chri'a r'0 sti nhi.lg h,r7p s6,
vA 115ng ih<ri ciruug t6 ring:
1r
€
I
'*
Hi".
!-
-+ :'
a;.,
\
. .i'
#
ra4
,/ [/IDN rr D r.. r. .\'iri
a'
pllQt la tic cuu J
2
sd -&-. 1o aguy€;r t6, thi no
,l
-l 1'1
\e
t
II 0NII ub, 2" .r'r,, if p iri sd ttgugin
phfit t* sd rtgltt:yirt tt\.
(.irrg ii6c r'&ng di.v iihdng phrii lA
rfui db ,lIo li ngu-vdn t6: chilng lian
.tfrr = 21i r l eZA+i : 23.89.
t6, thi p
olr-u Nlt I
YSy cd ti6u chuhn niro db bi6t vCri_ p nguy0n
t6, i14p ia nguy0r: lt5 iray Ii hr.rp sd'? 0 tl&.'r ttrh
hilh c6 ciriri trihri qurn hon sr.r vrii cAc sd Fec'nta.
'l'r'uuc lrdt trr co:
/ UnNg DE 3, .t'r-u p lri 2r) i t ltl ttliiug' s0
nqugin t6, ngoitt ra 2p{ 7 l:6 tluttg 81i -F 7. tfii lL{p
chia hlt cho 2p - I (ua dc-t tio llp 1ei lrot liop sr,).
I{0nh d0 i,:i,v e iro phirp phAt iridn ra hring- loa{
nhfr'ng sd }{ec-xen-rio hqrp s6: ching han J-/23
(chia hdt chr-i 47)" iUs: (chiit [rti[ , hir ]67) 11fil
(ehia hdt chr..26i-.j), i{lrg (chra h61 r'iro 3511), J^l:st
(cbia hdt cho ::l8li), l/zptt i'rlria irdt cho '1;1r)....
V6; li0lh de ll, neu'i;i lr l1v voilgr nLtrng
rhua clrurg rnilrh niri. i'lng ci.r vr) si, rrhirr,g idp
'hqp s6 voi p rrgrrl en tci. btli vi chua thung rrirth
nbi rirng eo vt) s,l so rgu,r-en ti, p sao cho
2p'l- l ngriv6n td, chrra cc,i d6rr r6u elu 2p + 1
pi:li c6 dar:g fiA'+ 7
Vidc x*c di nh cAc sd \'let:-xen-no hop sd hitin
nhi6n kii0ng tirir i'l biing vi€c -r{c dfnh cit'sd
l{ec-xen-rio ngurBn t6. C6 rndt. k6t quir nbi ti6ng,
nr6i tinr ra c*ch clAy khoirirg btln chuc, nfrrn biri
hai nhi. toan iroc Lucas r;i l-ehmer:
,/ DINH Lv LUCAS * LEHMER. Gitt thir p lir. mot
sd nguu|n td t6. Di'du ki€n r;hn ud. di db Mp
n11uy€.ntd Id: Mp ld toc cia sd hang thn p - 1
cia dfr.g Dn (n:1, ?,5,...), ddg.ndu rlaqc xdc dinh
b:)'i ed.c dtdts ki€n :
, 07 = 4, un+l : uA - 2l;hi n:1,2, 3,,,.
Vi khdng ttr6 bi6u diiii trr.rc ti6p vA rlcrn gi$n
uq euri n, nOrr ngu<ri ta {p dgng dinhlj'Lucas-
I-ehmer rhu sau: ,41p 1A u6c cfra up*l khi r,{
chi khi ltrf p ld uirc cira ,"S*1, trolg d6 d[y rn
(n: 1, 2,...) duqc x&c iiinh b$i: 11 = 4, ro41 lA
s6 clu trung phip ehia r3 * 2 cho Mp (eac bqn
dting klrAi r:i€nr ddng rlu d& chirng t6 dli6u d6).
Thdnh thiir trong r,i6c 6p dung dinh ly Lucas*
I-ehmcr, ta ehi ph&i binh ph*ong nhtng sd nh6
hon Mp, ldrn ph6p tru cho !, rbi ehia eho Mp,
ldy s6 du, r'A cr'v tidp tuc nhrt thd ! Di nhidn
kh6ng thb thuc hiEn cr6e ph6p [inh ni-r, bing
tay khi p dri lon: ilgay vrli Mrs:8191, b+n
ehS"rig n*n lirrn nhir v6y, tdt rh{t ban hly tra
mOt b&ug sd nguy0n t6 (thrrdng c6 Lrong b*.ng
ldgarit v*i 5 s6 thAp phir:), bg.tr s€ tlr{y ri"ng
.&fts lA sd r,rgu-v6n td I Nhurg du b1n c6 trong
tay birg ti tri6u s6 r:guy6n t6 clAu tiOn'crla
Baker-Grunberger (b&ng i6n nhdt hi{u nay
khoirg th6 i n thAnh sdch. e hl cd du<ri dang
rnicr'6finr. sd nguv0n td thu 6 1ri0u 1A p6000000 =
iU4 3$5 301), vA ban c6 lirb tiriir bltg tay trong
vrii irhrit s6 Mer:21,17!836,1?, 1hi b4n ch&ng bi6t
,XIg1 lA ngu-y0n tti lray lA irop s6, vd *h6c chin
ban kh6ug cd J rlintr fip dung b&ng ttr,y dinh
ly l-ucas*l.ehmer I Bic tidrr bdi (}-1e, s6ng trudc
I-ueas*Lehmer lrai Ihd i{i, {A ding Iy thuy6t
chi s6 db chirrrg nrir:h rairg ifsl lA ed nguy6n t6
r6 ltr s6 rrgu1,€n i6 ltl,r rririi dtuoe biSt thdi d6'
SLr ruril. hifu cic rnlv tlrih diQn lti dn giirp
r:lto r iQq 6p duirg 11ryc {idp tl in[: ji'LttcRs*Lehmer:
rgrrv siru klri ph6l rrritth ra rniL-v linh diQn ltr
(rrxy tird 1r0 ihu rrlrIl). rrgrrui ta tl1 thr] dintr
l_i irdri nri-y elro sd Mlot icri 3i chir s0), va
t h{ri rrirrg n(r lir hrr;i sii. ii hi Nikel v{ NoIl c6r:g
Ld sd J.l'?1Toi, c:ilc rriri ttrar: h?c 'l'ukerman vA
i-cliurcr ( vi'r l:" lrh;! [o:i r; iror; tinr ra dinh lf
iii r : con irl; r,hir to*r', iro r D.N. Lehmer (bi)),
lac giri i-,rinr llii ci gf1 . .i, rrgu-r 6n td nhi hon
Iil tr idu, rurit birn rrirrr l9r)9) cirirg tltng dinh lj
:11'dE ki6rn nghi0rn tr'*ln rnar"tlnL,1i{l tir {hE
ir6 thu ba, r'i r1A phi; tiur,g khoing {400 gio
rrr6y I
[Ii0n nn-v rrgrroi [:,. bi6t l5 sd rrgu;r€u tc \lec-
xeir-ro, d6 lA cAc sd Mp (r::g vtii p = 2,3, 5,7,
13, i7, il. 31, 61, Eti. li],, 1!7, 5?1, 607, 127!r,
221:1, 2281,32i7. 4?53,4i23, 9831-}, 9C4,i, 11213 (tlm
ra ndm igfiq), 193:17 (1inr ra n5:n 1971 b&i'I'uker-
mau) v[ 2!7d1. CAo sd iu.PiSz! tr& di d€u ltrrqc
ph6t hiQn nho mAy tilli di6n iri'.
Dfr sao nguoi ta cir rrg bi6t drroc rrhiEu sd
nguydn t6 Mec-ren-no hrrn ln sd ugu-y0n td
Fe€-rnu, vA hy vong tuy ci..r phln rdt m6ng marh
ring c6 v6 s6 s6 nguyen td Mec-xen-no. Cd thd:
nguoi la cho riug n6u &lp l& s6 ngu,v6n t6, thi
Mso e irrrg II s6 lgu_v0n td : di6u 116 ditng chtt ,tr
sd nguy0r.r ,t6 ,ilIec-ren-no deu tirlu. Xoi:g udni
1953 trVheeler dring rlinli i1' l-ucas-Ltlhrner t16n
rnhy rinh tliQn tiil de cirung t6 rirng sd
Mf,lc :28191 * I (cd 246r; cbfr sd) ia hqp sd I
N[nr 1957 tguoi ta cln chtln$ rninh tluge ri"ng
sd Mp,, ctriu hit clro 1768 (2" - l) r'A s6 lvls*
.nio f,6t *no 120 (21e - l). llifn gi6 ngudi ia
h1, vqng ring tdt c[ c6c e6 (io. Ql. q2...., trong
dd 1 qo :2, {n*t: ?qo -- 1 (rr :0, l. 2,...) tIEu
lA sri" rrguydrr l6 ! Di6u d6 dii ng cho go - 2,
el:2" - I = 3, eZ:.2*'- i = 7, g3:2' - l=-12i,
i+=zr'7 - t:,l/irz. D6n s{i qs:iut:2"-1 c6 hcvn
io37 chft sd, rrgutii ta darih ehiu, kh6ng th&
kitsrn nghiQm nbi n6 lI ng,rysn 16 hay lA hqp sd !
C{c ban c6 th6_ h6i ; vi sao irhi di tlm sci
nguy€n t6 16lr nhflt, nguoi ts ehi ehir y cl6n trAc
e
&
I
s6 Mec-xcn*n() m{ khdng iinr ciic s6 dr.rng khAr:'i
D6 ki i'i ba rrguv6n nhilrr :
1) DE chirng t6 rang sd N l* nguyBn t6. vii
rnit nguydn t6e, ngriri ta plrfri clting c6ch sirrg
't-ra-t0-xlen, aoi cfrch khAc, ptri i k ibrn rghiem
rirng N khOng co u6c nguydn td p -{ f/ H xt O"g
nr,ri cl6n sn t6n hi:rn, nhtr;rg viQc IA,ni ni-r- khOng
thB thqc hi€n tlLrgc khi .\ c6 hon hai chuc ch*
s6 : c\o cldrr na1-, ngttoi ta chua bi6t dtroc tdl
cd c:lc s6 ngtiyirii td c,r tri rnrrdi chu s6 trri'
xu6r'rg. r
Cudi tlrd kj; 18, rrhe lti6 ri lro,: l.cibrriz ili thu rrg
rninlr :
,/ DINrt t.{ LEiBNIZ : #! s0 tv rthitrtp> 2 l{r
nguyen tb, ttibu ?.'i{n cI.rr ud dil Id sd (p - 2)l *' I
chirt hil t'ho p
Dinh lf Leibniz khdng pliii Id mdt c6ng i:u
c<i hi6u iu. c clE phht hi6n nlrirng s6 ngu-v6n to
l6n : quA viv, ngui.ri ta cld bi6t tdl cA cac sd
nguvcn tO nfril tr,ri, tO8 ( birng Baker-Grtrnberger);
d6i v6i cilc sd,v>10", ild Ap dung d.ilrh li't.ei-
bniz, la phii .tinh (lg * 2) l, c(r tii} coi r"ing
A'- 2: n l lu" vi [;, t.,
hoin r:hi nh lii. thi it plrii ,'t, v6 sd sd horln
,'hinh li'l
Bi-r- gio-. xi rr rroi th6rn vdi cric bq.n vd tlnh
(don giln> cira dli sd l{er-xen-nr-r. YA m4t
1f thuy6t thuiii: ttil;. drrorrg rrhien con c6 rrhi€u
dIv -s6 ilon gif,rr iro'ir, Chri ng ha rr diir'
n2 -1 1 (6:: l), 1, ?,...)
i)il nly chria vo so nirirng hrrp s0. ch,rng
han cAc sd hnng rlrng vtri l iAn cirng birirg 1.2, d
ii,5,7,8, 9'(trri trLtdng liup n = i, 2), huic T =2mz
vdi rrr ) l. biir r i kfri tld n" -F 1: {rrr" * I :
Ln,2 I i)'- lnr?= (irirs - 2tn + l):*.t2m2 - 2m +l). +;
Nhirrrg dir.t :1-v citrg chtia Irhi0rr sd rlguv€rr 16, 'r!-
+l:17.t,2+l=37,
iri?r I=r)57,')(P+7
n6i c6ch khne ; (tr - 2)l cO hon 7.1(t8 ch-ir sd, co
lE co n liu lt'irrr r6c m61; tinh tli$n tu m6i c6
khi nlng l[rn vi€c vtii cic s6 niy !
Nhu vA.r- chi c6 viQc ki6nr nghidm c6c sd
Mec-xen-no, nhd dinh I,v Lucas-Lehmer, rndi
thuc hi6n ilur:c ther: khi n:rng cfia mAv tinh
di{n tir, may cAng ( tn+trh > thi cAng co rliEu
kiBn tim ra s6 ngur-,0n id Mee-xcn-no ltrn hrrrr
2) 56 l{ec-xen-rru c6 dang khdr clon gi&n : nir
Ii nrQt liiy thira cira 2 trri di 1' Eidu rrAv r'{t
thuAn tiQn cho cAc ntiy tinh didn t&: trhtlirg linr
vigc vdi hd d6nr theo co sri 2 (\'i rnach di6n tri
hai lrang thiri: hr>[c. ttdlg, lroiic mir.;.
3)'l'rong l-r: thtrydt s6, s6 tr'iec-ren-ncr cd Iidu
quan vdi c6,c s6 hodn chinlt. Mdt s6 tlt nhi0r: I'
ituoc goi Id hoAn chinh n6u n6 bing tbng t{t crir
eftc u6c tu nlri€n nh6 hnn P cria sd P' C.Ac srj
hoirr chinh dl['dugc guan tAm ngal' tu thr-ri
0-c;lit i' theo ng6n ngir hi0n dqri. 0-rrlit il1, chi'
ra ring n6u i"Ip 1e mf;i s6 NJec-ren-ntr, thi 2P-1 \lp
ld m6t s6 hodn chinh (ch:rn). 'Ihd h-i 18,. 0-lr:
t'tI cliitng rninh chi,it chO: t{.t ci cic sd hcrdn c}rintr
chin ld trlt ci chc sd trOn.'IhAnh thri'bi6t dutvc
bao nhi6u s6 trguyAn td Mec-:,:en-l<r, ta bi€t ducrc
ching 6y sd hoan chinh chirrr. Mtrt khac cho d6'rr
na--v, rgudi ia v6n chua bi6t cd hay kh6ng, co
sd hoirn chinh 16. Co cli0u tlty vi ia ngurli ta
riir chrlng . rriinh ring : l6u c6 it nli[t InOt s6
4
Dly nij chria vtr s$ t6 ngtry€n td chdttg'J
Ngu'di 1a nghi rriru vi]-, nhrrng chua chtirtg minh
rrbi I Ngndi t.r cfing r:hra bi6t ur6,t diEn kiQn
eln r.A dri, cd Itil:u lrrr', chr, bidt ;r' + I lA mQt
s6 nguvdn t6. tlq,c biQt eho, ph6p phat hi€rr nhiing
si rrguli rr t6 Lru dang n' -l I.
Con e i; m6t ioai dir,v s6 riit don gifin, rni cae .
ban rlBu bidt: cAc c{p si cQng
a -t nd. (n : 0. I. 2....).
C[c dly s6 tr1 nhi€n, rtl-r s6 chirl, cliy s6 ii',. .,
d6u ln cf,p s6 cQng. C6 thB chung rninh d6 dri,rrg
ririrg bdt ctl cdp sti cdng n{o (v6i a vA d }i
uhir'ng s6 tr,r nhi6n) i16u chirrt v0 s6 nhfr'ng hqp
s6. Nguoi h khdng bi€t rn6t diEu ki$rr ciin vd
dir, cd iri6u itrc, cho phdp xiic dinh rnQt s6 hatrg
a* nd, clu cria mOt ctip s6 cqng cq th6, le s6
nguy6n 16. Thd nhutrg nhl-r 1837, nhir to{n hr-rc
Di-rieh-trd ilir chirng mitih l
I)lNIi i-lrt Di-HIO-Lll. NdIl a ud cl la hai sd /u
nhiin nguytln tii cing nlruu, t]1i \ cdP sd cQng
a'l- ttrl (n:0. 1,2,..-) ehna o6 sdsd ngtgAn t6.
Ii€t quii nd,v ilunc <lanh gi6 lr\ rlQt thirnh tuu >
ruc ro cfrl 1i,' thuv6l sd. Ngudi ta cang th6m
Trry kdt quii ph6t bi6ri rlorr giir n, nhung dB
chirng minh, Di-rich-16 pli:ri vrirl dr.rn$ cl6rr nhr'rng
lii{n thuc khir cao si€u, & lrinh ato tr6n dai hoc.
I)6i vdi vti cip sd cdng cr1 th6' phtip chrtng
minh c6 thb alr-rn giin hon : chitng ltan cic ban
cri tir6 clirtrrg minlr et6 dipg'cac dE-r' sri 4A -l 3
",= (t)", / ro8 \lu - 7.108
l;/ >ru
chine hau:
12+1:2,'12 l=i,42
luz1l=t01.t42+1=1u7,
= 401,...
thia voi kdt qui 5y hon, vi bi1' gid iiI gin a*\
h6t thii k(, 20, ngoei cric c5p s6 c0ng n6i tri'n, ;flf,
trong c6c ctiy sd don gi:in thuong e-[p, nguoi !4d
ta chua cirri'ng rninh n6i c6 r'r'rdt diy sd niro il 6
chita rd s6 sdi nguv6n tri I
(& = 0, 1, 2,..1) h:ry 6ft *; (li = 0, l. 2,...) chira
vO sd stj nguvdn 1fl. vi vfry Irirm 19.18, nhir loan
hoc Selberg chung nri nh lai din h l-'i" I)i-rich-16
blrrg phuorry phfLp srr cflp: chch chrrng minlr [v
c16 niang lai vinh dr.r l6n cho Selberg vA nl&
tllu mOl phrong hu6ng nghiOn cuu rniri trong
lf thrry6t s6 : chf'ns minh (cic, kdt qui rnd,i
vi cir) bing phu<rng ph6p so' c{p. 6oi lA ( so
cdp D. nbung r'(ri trinh c16 phAt triEn lr-ian
hgc hiQn nay, c6c ban cin hiilu li: dring cAc
.
kii:n thtrc ioiru hgc & ba hay- bdn n{tn cllu 6
clai, hoc.
l'orn lai, cric brrrr tlris lridrr girr chi ddi .vdi
c6c sd Mec-xett-tto. ta m<ji c6 nrdt ti6u chuhn c6
hi6u luc cho ph6p xAc <linh s6 dy ld nguvdn t6
har-, kh0ng nguien td : clinh l-{' l,ncas * Lehrner.
Vi tAy ,,6o t.o,rg rn6t vAi nim tdi c6 ai d6 tim'
r:r sd irgurdn t6 l6n ho-n, thi kh6ng ph&i lA nha
tion tl:i. cic birn c6 thb ti6n doan Ct6 lai td mQt
sri ngutin tu )[ce-ren-ttu ]
{.
ffi
KY
AU
THI VC D/CH VE IoAN c&,q CAC
IN
kJi thi r.6 clich vd todn cho hoc sinh cAclrrrirng
phb ih6ng & Lierr Xb thn thu 14.
Du thi c6 158 hoc sinh rria ciri lrip 8, 9, 10.
'Nhti'ng hoc sinh ni_r gdm hai rleii tuong: lror
sinh cl:rt giiii trong cAc kr'thi vd dich 6.c{c
nucrc cdng hoa vA nhirnq hoc sinh tiuoc c:1p bing
loai I vir II 6'kj. thi vt) dich lin th* 13. -\godi
la thinh phd Xanrtdp i[uoc cri ri6ng m6t dOi
gdm ba hgc sinh clu thi.
Ngu<ri llnh clati ba n gi6m khio h-i' thi ld viOn
si CdnmdgOrOp,
C{c thi sinh plr6i lirm bdi thi vi6t trorrI hai
ngn5r 18 r'4 20 1lrlng trr. mr-ri ugr,r lanr 4 bdi toarr
trong 5 gid. cli6m tdi da trong m6i ngdv la lJ(}.
DiSm thi cira th{ sinh ilat dtrcrc trong hai ngiy
d[ucc cQng lai vA dugc cip hing theo qui clinh:
Bing loai I cho thi .sinh drrt tri 57 d6n 60 dibm,
loai I1 tu 40 d6n 56 cli6m vd loai III tu. 30 d6n
39 etiBrn.
NgAy 22 thAng ,1, tai gian phong ilai 16 Dovdxa
6 thdnh ph6 Xaratdp dd tb ch'irc long tlong 16
b6 rnac k)t thi v6. dich lAn thu 14 vA trao gi&i
thrr&ng cho nhtno hoc sinh duqc giAi. Hon nr?a
s6 hoe sinh du' thi dd dlugc c{p br'rng, trong d6
loai I c6 7, Ioai II c6 35 vir loai tII c6 41 hoc
sinh. 40 hoc sinh iluoc cdp si{v khcn 56 oon
iai tluoc cf,p hing hgc sinh 4lt thi ud diclt htn
bang ldrt tlttr '14" Chi c6 rr.rgt hoc sinh d4t di6nr
t6i da id M. (iaixinxki. Ba hoc slrih -khic LI
IRUOI,/G PHb THONG i IIEN XE T,AX/ TI-IU' 14
DAN(] HAI NAU
I
ttAn* Oip ki ni6m lrln thL? 110 ngly stnh V.
L€-uin. tri'ngAy i6 c16n 23 tharg 4 niinr
1980, tai thdnh pbd Xarat6p dir ti6rr 'hA nh
N.Goribecrcr, D.Bur.al,6 vA \I. Triirenc0 turrg dat
59 di6m." N. Go-rinbeclc.r ilucrc'tf,ng darih hicu
ngudi con gai gtiji nhdf vi drroc tEng nhirng phrln
thu3ng ri0ng. Hoe.sinh ,\. RagiobarOp ba nim
liOn ti6p rluoc cdp bing loai 1, va tai kj thi
qu6c t6 tB chr'rc tai LrrAn DOn, hec sinh n{v
clng dugc cip bdng loai I, Diic biQt doi hoc
sinh cria thdnb phd tt5ng cai, hai hoc sinh dtroc
c6p bing loai II vA hqa sinh con lai dugc c5p
bi trg loa i III.
Duoi <tay l{ rJ0 thi trong hni ng*1', r
NC,{,Y ra{r nrr,{r'
Ldf, I
/ 1.. Vi6L libn tidp cdrc sei c6 hai cht s6 tir 19
<l6n 80, ta iluoc s6 19202;...?980. X6t xem s6 nay
cdr chia hdt cho 1980 hry kh6ng ? (6 di$m)
2. Canir .4.8 efia hinh r.u6ng ABCD ilrrgc chia
lirn n tlnan sao cho tbng dO dAi c6c iloan v6i
chi sd chin bang t6ng d6 dii edc doan v6i chi
sd 16. Qua eic diBm chia ngurii ta k6 c6c dudng
th&ng song song vdi canh CD chia .hinh vu6ng
thil1h n d&i, m5i dli bi tiuong eh6o hiDh vu6ng
chia thAnh hai phiin: b0n trSi vrl b6n phii. Cht?ng
minh ring tbng digrr tich c6c phCn voi chi s6
chin bhng tbng di6n tich ehc phAn v6i chi s6
16. (6 tli6rn)
3. Nguoi ta ciin phii tlua ldn qui dqo cfia lhu
vt trg tr4nr ( Chdo muqlg ),.Cdc hing v4n chuyBn