Abstrct Pell equto s mportt reserch object elemetry umber theory of defte equto ts form s smple, but t s rch ture My umber theory problems ce trsforme

The solvblty of egtve Pell equto Jq Wg, Lde C Metor: Xgxue J Bejg Ntol Dy School No66 Yuqu Rod Hd Dst Bejg Ch PC December 8, 013 1 / 4 Pge - 38

Abstrct Pell equto s mportt reserch object elemetry umber theory of defte equto ts form s smple, but t s rch ture My umber theory problems ce trsformed to the problem of Pell equto s solvblty However, the prevous methods determg the Pell equto s solvblty re sophstcted for clculto, whch leds to the lck of effcecy Ths pper gves ew d more wdely used methods to determe the solvblty of Pell equto, cludg severl ecessry codtos, suffcet codtos d ecessry d suffcet codtos Keywords Pell equto, Solvblty, Cotued frcto expso, Perodc / 4 Pge - 39

1 Itroducto 11 Bckgroud d Motvto I studyg the theory of cotued frcto expsos, we otced the expso of D -type of rrtol umbers were perodc, wth ts prty of perod closely relted to the solvblty of egtve Pell equto Its perod s odd f d oly f egtve Pell equto y 1s solvble It s well kow tht the postve Pell equto, (11) x Dy 1, hs ftely my solutos ( xy, ) wheres the egtve Pell equto, (1) y 1, does ot lwys hve soluto Heceforth, D wll deote postve squre-free teger, d soluto refer to postve tegrl soluto The Pell equto s oe of the oldest Dophte equto tht hs terested mthemtcs ll over the world for more th 1000 yers The most essetl problem of these two Pell equtos s determg ther solvblty d how to fd out the soluto to the Pell equtos s quckly d cocsely s possble Archmedes fmous cttle problem ws ctully problem bout the solvblty of the postve Pell equto The soluto to postve Pell equto s sd to be gve by the Id mthemtc Brghmgupt 6th cetury The, the 17th cetury, Lgrge dvced hs predecessor s theorem d gve the soluto to postve Pell equto by cotued frcto expso theory Hs result ws tht the postve Pell equto s solvble for y gve D, d, the solvblty of the egtve Pell equto s determed by the prty of the perod of D s cotued frcto expso The equto s solvble f d oly f the perod s odd umber The cotued frcto expso s wy to costruct solutos Not oly for huge D my t becomes extremely tme cosumg, eve for some smll Ds, the soluto to the Pell equto s huge For exmple, for D=61, we hve ( xy, ) (9718,3805) for egtve Pell equto The solutos of whch would hve huge tme expese Accordg to [10] d [11], ts rug tme s pprochgo D Besdes Lgrge, s [7] shows, my other mthemtcs hve mde mor cotrbutos to the problem reltg the solvblty of the Pell equto The most populr method tht hs mde gret progress tody s reltg the solvblty of egtve Pell equto to the solvblty of other Dophte equtos I 1986, Keeth Hrdy d KeethSWllms proved tht d oly f D s the sum of d 3 / 4 y 1s solvble f b, where b,,d s odd umber such tht the Dophte equto bv VW bw 1of V,W s solvble The method bove s put to use o [15], whch mes tht for y gve D we c determe whether y 1s solvble or ot Of course, the clculto must be wth the lmtto of computtol power AGrytczuk, F Luc d M W ójtowcz [1] proved tht the egtve Pell equto y 1s solvble f d oly f there exst prmtve Pythgore trple ( A, B, C ) (e A, B, C re postve tegers stsfyg tegers bsuch, tht D b d A bb 1 A B C d gcd ( AB, ) 1 ) d postve Pge - 40

Of gretest vlue here s the reducto of the coeffcet of the egtve Pell equto, whch my reduce the tme complexty determg the solvblty Aother pproch to ths problem volves plcg codtos o the modulr resdues of D whch gurtee tht (1) s solvble or usolvble Ths pproch ws tted by Legedre 1785 Drchlet[13] observed tht f D pq wth pq1(mod4) d p q p = 1(where mes tht p s the power 4 resdue modulo q ), the (1) s q p 4 q 4 4 solvble For D p1 p p3 pn, To[8] obted qudrtc resdue crter mog the p, whch whe they held would gurtee (1) s solvble Explct modulr resdue codtos for prtculr clsses of D re stll beg foud, eg Kpl[6], Pumplü[14] However, for some huge D s, the prevous results re lck of effcecy Ths motvtes us to promote the prevous results d gve qucker lgorthm to determe the solvblty of (1) All the methods dscussed ths essy re ot oly cpble of determg the solvblty of egtve Pell equto y 1, but c lso be used to solve my problems relevt to the geerl Pell equto For stce, they ce used to determe the prty of the perod of the cotued frcto expso of the rrtol umber D etc 1 The cotet I secto, we gve the prelmres, cludg severl mportt Lemms Pell equto d the propertes of qudrtc resdue We lso gve some ottos tht re used the proof I secto 31, we gve ecessry codto for y 1to be solvble I secto 3, we gve some suffcet codtos for the solvblty of y 1 We gve lgorthm to determe the solvblty of y 1for some prtculr D s The, we defe fucto, whch ce used to preset suffcet codto to the solvblty of y 1bse o the lgorthm We lso gve seres of corollres I secto 33, we frst rrve t ecessry d suffcet codto for determg the solvblty of y 1d we gve severl D s by the codto for whch y 1s solvble It forms theorem 6 By theorem 6, we gve out severl exmples to show tht there re my types of solvble egtve Pell equto wth qudrtc form D Secodly, theorem 7 gves suffcet d ecessry codto for determg the solvblty of y 1whe D Y I secto 4, we look bck o our reserch, do some umercl expermets d gve severl types of solvble egtve Pell equtos I the ed, we gve severl comprsos wth the prevous results I secto 5, we gve the cocluso I secto 6, we gve our pls for future work Prelmry Lemm1Assume tht p s odd prme umber, -1 s the qudrtc resdue of p f d oly f p 1 (mod4) Tke odd prme p, we defe the fucto wth the teger vrble d 4 / 4 Pge - 41

1, whe d s the qudrtc resdue modulo p, d 1, whe d s't the qudrtc resdue modulo p, p 0, whe d p d We cll the Legedre symbol of d modulo p p Therefore, Lemm1 ce rewrtte s 1 p 1(mod 4) 1 p LemmAssume tht D s postve squre-free teger, x Dy 1s solvble Its geerl soluto ( x, y) ce gve by ts lest postve soluto ( x0, y0) through (1) x y D x y D The followg recurso formul ce prove by (1) 1 ( 0 0 ), x x x x x1 x0 1 1 0 Lemm3Assume tht D s postve squre-free teger, f y 1s solvble, the ts geerl soluto ( x, y) ce gve by ts lest postve soluto ( x0, y0) through y y 1 ( 0 0), m Lemm4Assume tht p s odd prme umber, If 1, r m (mod p) hs oly two p solutos r 1, r (0 r1 r p), wth r 1 r p Lemm5Assume tht m,, z ( m, ) 1, m z, the there exst uv,, such tht m u,, v, where( uv, ) 1 The proof of the bove lemms s vlble [] 3 M Work 31 Necessry codtos Theorem 1If y 1s solvble, the D s ot dvsble by 4 d D does t hve prme fctor cogruet to 3modulo 4 Proof If prme p such tht p 3 (mod4) d p D, tkg y y 1 (mod p) Ths mes tht 1 s the qudrtc resdue of p, cotrdctg lemm 1 If 4 D, tkg y 1modulo 4 gves 1modulo p gves 5 / 4 Pge - 4

y 1 (mod 4) However, y 0,1 (mod 4), ledg to cotrdcto To coclude, t s ecessry codto for D f y 1s solvble We gve the followg defto of three sets for smplcty Defto 1 Y : p p 1 (mod4), p s prme umber}, we cll t set of excellet prme umbers s Y : x x p, p Ys,,, ( 1), we cll t set of excellet odd 1 1 umbers U : x x p, p Ys,,, we cll t set of excellet eve umbers 1 By defto 1, theorem 1 ce rewrtte s If y 1s solvble, the DY or D U Remrk The ecessry codto o D gve [1] s equvlet to theorem 1, by Fermt s theorem o sum of two squres 3 Suffcet codtos Theorem If DY or D U, the the geerl soluto( x, y ) to x Dy 1stsfes x, y Proof Becuse ( x, y ) s soluto to x Dy 1, we obt (31) x Dy 1 If D Y, tkg(31) modulo 4 gves x Dy x y 1 (mod4) Sce for y teger m, m 0,1(mod 4), we obt x, y If DU tkg (31) modulo 4 gves x Dy x y 1 (mod 4) Sce for y teger m, m 0,1(mod 4), we obt x, y Theorem 3 y 1 s solvble f () D ; () D p, p Y, ; s p () D p q, p, q Ys,, 1 ; q (v) D p, p 5 (mod8),, where p s prme Proof ()For the Pell equto x y 1, t s esy to verfy tht ( xy, ) (1,1) s soluto to 6 / 4 Pge - 43

the equto () Applyg Theorem, we obt the lest postve soluto ( x0, y0) to x Dy 1stsfyg x, y (3) Assumg y0, k k, we rewrte x 0 0 0 Dy0 1 x 1 x 1 s 0 0 x0 1 x0 1 x0 1 x0 1 Let m,, from x 0, we obt, N x 1 x 1 Becuse 0 = 0 +1, we obt x0 1 x0 1 (, ) =1 Nmely, ( m, ) 1 The (3)s equvlet to m p k Sce( m, ) 1, p s prme umber, by lemm 5, we c get two possble cses for m, Dk m u, m p u, (1) () p v, v, I both cses, ( uv, ) 1 I cse (1), p u Icse (), p v x0 1 x0 1 I cse (1), s m 1, we obt u p v 1 Nmely, ( uv, ) s soluto to x p y 1 Recllg x 0 Dy0 1, we obt x 0 Dy0 1 1, therefore x 0 1Becuse x 0, we obt x 0 3 Therefore, x0 1 x0 x0 u m x0 x0 Hece, ( uv, ) s soluto to x p y 1d u x0 However, ( x0, y0)s the lest postve soluto to x p y 1, ledg to cotrdcto m p u, Therefore, cse () holds, whch mes tht, the v, p u v m 1 It shows we fd soluto ( uv, ) to p x y u v 1, here, x0 1, p x0 1, 7 / 4 Pge - 44

()Applyg Theorem, we obt the lest postve soluto ( x0, y0) to x Dy 1stsfyg x, y (33) Assume tht y0, k k, we rewrte 0 0 x0 Dy0 1 x 1 x 1 0 0 8 / 4 s x0 1 x0 1 x0 1 x0 1 Let m,, from x 0, we obt, x 1 x 1 Becuse 0 = 0 +1, we obt x0 1 x0 1 (, ) =1 Nmely, ( m, ) 1 The (33)s equvlet to m p q k Becuse ( m, ) 1, pqre, prme umbers, by lemm 5,we c get 4 possble cses for m, b m p q u, m p u, m q u, m u, (1) () (3) b (4) v, q v, p v, p q v, ( uv, ) 1 holds ll cses p v, q v holds cse (1); p v, q u holds cse (); p u, q v holds cse (3); p u, q u holds cse(4) (34) Sce, we obt 1 1 p u ( p u) b I cse (), from p u q v m 1, we obt 1 b p u pq v p p p Tkg (34) modulo q gves 1 However, 1, ledg to cotrdcto q q b Icse (3), q u p v m 1, we obt (35) Sce, we obt b 1 pq u p v p 1 1 p v ( p v) p Tkg (35) modulo q gves 1 q However, p 1 p 1 ( 1) 1, q q q ledg to cotrdcto I cse (4), u p q v 1 Nmely, we obt soluto ( uv, ) to x Dk p q y 1, Pge - 45

x0 1 d u m x0, whch mes tht ( uv, ) s smller soluto th ( x0, y 0) However, ( x0, y0) s the lest postve soluto to x Dy 1, cotrdcto m p q u, I cocluso, cse(1) holds, whch mes tht Therefore, v, p q u v 1 So we hve foud soluto ( uv, ) to p q x y 1 m x 1 x 1 p q p q 0 0 ( u, v) (, ) (, ) (v)applyg Theorem, we obt the lest postve soluto ( x0, y0) to x Dy 1stsfyg x, y (36) Assume tht y0, k k, we rewrte 0 0 x0 Dy0 1 x 1 x 1 0 0 9 / 4 s x0 1 x0 1 x0 1 x0 1 Let m,, from x 0, we obt, x 1 x 1 Becuse 0 = 0 +1, we obt x0 1 x0 1 (, ) =1 Nmely ( m, ) 1 The (36) s equvlet to m p k Becuse( m), =1, by lemm 5, we obt four possble cses m p u, m p u, m u, m u, (1) () (3) (4) v, v, p v, p v, I ll cses, ( uv, ) 1 I cse(1) p v, v I cse(), p v, u I cse(3), p u, v I cse (4), p u, u The sme procedure the proof of theorem 13 ce dpted to elmte cse (),(3),(4)To coclude, cse (1) holds, mely m p u, v, Therefore, p u v 1 So we hve soluto ( uv, ) to p x y 1, d Dk m x 1 x 1 0 0 ( uv, ) (, ) (, ) p 4p We hope to geerlze Theorem 3 to cse whch D hs more prme fctors By cosderg the smlrtes the tretmet of dfferet cses theorem 3, we look for fucto to determe Pge - 46

the solvblty of y 1, where D Y For exmple, let D 513 17, we frst brek D to A, B, such tht ( A, B) 1, AB D, A D, B D There re the followg three cses totlly Cse (1): A 5, B13 17, We kow tht Implyg for cse we hve 5 13 3 1 13 5 5 m 5 u, 13 17 v, 5u 13 17 v 1 Multplyg the bove equto by 5, we obt 5 u 5 13 17 v 5 Tkg the prevous equto modulo 13 gves, However, ledg to cotrdcto Cse (): We kow tht Cse (3): We kow tht A17, B 5 13 5 17 1; 17 5 5 A13, B 5 17 5 13 3 1; 13 5 5 5 1 13 5 1, 13 I cse () d (3), we c dpt the sme procedure cse (1) Therefore, 5 13 17 x y 1 s solvble, we check D o [15], t shows513 17 x y 1s solvble g : \ 0,1, Defto Theorem 4 For y x \ 0,1 x 1, x, 10 / 4 x p, 1 p f there exsts such tht, gx ( ) 1 f x s squre,, g ( x ) x s squre Pge - 47

Proof If x s squre, the Therefore, g( x) x s squre g( x) x 1 x x m k bj j, 1 j1 If x s ot squre, ssumg tht x p q where, bj, m, k,, b j The m m k m k j b j ( ) j j 1 1 j1 1 j1 g x x p ( p q ) p q Sce for y, s eve teger, g( x) x s squre For D Y, we c try to determe the solvblty of y Defto 3 f : N\ 0,1 0,1, f( D) s the output of WCJ lgorthm 1by the followg lgorthm ---------------------------------------------------------------------------------------------------------------------- WCJ lgorthm ---------------------------------------------------------------------------------------------------------------------- 1: Iput D Y ; f( D) 0 ; : for ech ( A, B) 1, AB D, A D, B D { 3: possble = true; 4: for ech prme fctor p of B { 5: g( A) f 1 { p 6: possble = flse; 7: brek; 8: } 9: } 10: If possble == true 11: Retur; 1: } 13: retur f( D) 1; -------------------------------------------------------------------------------------------------------------------- Expressg the result by the fucto, we obt theorem 5 Theorem 5 If DY d f( D ) =1, the y 1s solvble Proof For D Y, ssume tht ts prme fctorzto s D p Therefore, there re kds of wys to brek D to AB,, such tht( A, B) 1, AB D, A D, B D For every ( AB, ), exme defte equto Ax By 1 Multplyg the bove equto by ga ( ), we obt ( g( A) A) x By g( A) By Theorem 4, we obt g( A) A s squre Therefore, for y prme fctor p of D, we obt 1 11 / 4 Pge - 48

( g( A) A) x g( A) (mod p) Nmely, g( A) =1 p g( A) For every prme fctor p of B, we clculte the vlue of If oe of the p to 1, the Ax By 1s usolvble g( A) p s equls Sce f( D ) = 1 mes tht ll the kds of wys to brek D to AB,, ( AB, ) 1, A D, B D, Ax By 1s usolvble If ( A, B) (1, D), dpt the sme procedure the proof of theorem 3, we wll fd cotrdcto Therefore, the fl cse holds, t derves tht y Remrk1 Ths method my ot elmte some cses for gve D Y Eg D 5 13 17, we get three possble cses 1s solvble Cse (1) A13, B 5 17, Cse () A17, B 5 13 ; 17 4 1; 13 13 13 Cse (3) A 5, B13 17 ; I cse (1), oe of the vlue of the Legedre symbols s 1 Therefore, f( D) 0 However, we c check D 5 13 17 o [15] d ctully solvble 5 13 17x y 1 s Remrk The prty of the power of ech prme fctor p of D s the oly mportt fctor the lgorthm Therefore, we c reduce the power of eormous D to 1or Ths mes tht we c costruct huge possble D from smll D wth f( D) 1 By the m de of the lgorthm, we c gve the followg corollres: Corollry 1 Gve m k j j, where, bj, 1 j1 D p q m, k,, b j, d f( D) 1, the for y c, dj, m, k, c, d j, m k c d j j, we 1 j1 E p q hve f( E) 1 Nmely, Ex y 1 s solvble p Corollry Assume tht p, q Ys, 1 A { x x q (mod4 p), x s prme} q p,, 1 p A, pp p p x y 1 1 s solvble 1 / 4 Pge - 49

Corollry 3Assume tht p1, p,, p re prmes whch cogruet to 13 modulo 0, s odd umber, the 5 p1p px y 1 s solvble Corollry 4Assume tht p1, p,, p re prmes whch cogruet to 17 modulo 0, s odd umber, the 5 p1p px y 1 s solvble Corollry 5Assume tht p1, p,, p re prmes whch cogruet to 5 modulo 5, s odd umber, the 5 p1p px y 1 s solvble Corollry 6Assume tht p1, p,, p re prmes whch cogruet to 1 modulo 5, s odd umber, the 5 p1p px y 1 s solvble Corollry 7Assume tht p1, p,, p re prmes whch cogruet to 33 modulo 5, s odd umber, the 5 p1p px y 1 s solvble Corollry 8Assume tht p1, p,, p re prmes whch cogruet to 37 modulo 5, s odd umber, the 5 p1p px y 1 s solvble Corollry 9Assume tht p1, p,, p re prmes whch cogruet to 41 modulo 5, s odd umber, the 5 p1p px y 1 s solvble Corollry 10Assume tht p1, p,, p re prmes whch cogruet to 45 modulo5, s odd umber, the 5 p1p px y 1 s solvble Corollry 11Assume tht p, q, r1, r,, rm, s1, s,, sk re ll excellet prme p umbers, 1 q p q, 1, 1 1,,, m, j1,,, k, d b re both r s j odd umbers, the p q rr r s s s x y s solvble 1 m 1 k 1 Corollry 1 Assume tht p, p, p, p, p, p, p, p, p,, 1 1,1 1, 1, m,1,, m 1 p, p, p re ll excellet prme umbers,,1,, m 13 / 4 Pge - 50

p p p p 1,, j {1,,3,, }, j, = 1 p, 1,,,, j p,1 p, p, m 1,,, re odd umbers, the s solvble m m m p p1, p, p, x y 1 1 1 1 1 1 33 Suffcet d ecessry codtos We observe tht for gve soluto ( xy, ) to egtve Pell equto, we c fd D for whch y 1 s solvble Therefore, by gog through ll solutos ( xy, ) to y 1, we c obt every D tht for whch y 1s solvble Assume tht x s teger d for y prme fctor p of x, p Y The solutos to the s cogruece equto r 1 (mod x ) re r r r r x,( 0 r r r x ) x,1, x,,, x, x (mod ) x,1 x, x, x By lemm 4, we c obt x M x, Deote r x, 1 by M x,, 1,,, x, the x Theorem6 y 1s solvble f d oly f ( x, such tht for y prme fctor p of x, we hve p 1 (mod 4), d D = x k r x, k M x, s sutble for {1,,, x } d turl umber k ) or (there s k, such tht k N d Dk 1) Proof Becuse the y 1s solvble, ssume tht ( x0, y0) s soluto to y It ce rewrtte s D 0 x0 0 y0 1 1 Sce D, x 0 0 x cogruet to 1 modulo 4 If x0 1, ssume tht the solutos to the cogruece equto re y 1, y 1, by lemm1, ll prme fctors of r r, r,, r (mod x ),0 r r r x x0,1 x0, x0, x 0 x,1 x, x, 0 x By lemm 4 d the Chese remder theorem, we c obt x The there exst 1,,, x, d k, such tht 14 / 4 r kx y Deote x0, 0 0 r x 0, x0 1 Pge - 51

by M x 0, The y ( r kx ) 1 D x k r k M 0 1 x0, 0 0 x0, x0, x0 x0 If x0 1, the D y 0 1 There s k, such tht Dk 1 x x0, Coversely, whe D x0 k r x 0, k M x 0,, deote y kx r to y 1 The Hece, Whe D k y = ( kx r ) 0 0 x0, ( x k r k M ) x ( kx r ) 0 x0, x0, 0 0 x0, r 1 M x r x r x0, x0, 0 x0, 0 x0, x0 1 y 1s solvble x 1, 1, let d substtute t to y 1, y k, y D k k 1 k 1 Therefore, y 1s solvble 15 / 4 0 x0, d substtute t, By theorem 6, we c costruct fte sequeces of umbers such tht ll terms, represets D for whch For exmple, y 1 solvble x 1 r, r M1, M 5 7,18,13 13 70,99 9,58 17 38,51 5,18 9 41,800,701 Tble 1 the qudrtc form D by theorem 6 G(5 ;7; k) 5k 14k, G(5 ;18; k) 5k 36k 13, G(13 ;70; k) 169k 140k 9, G(13 ;99; k) 169k 198k 58, G(17 ;38; k) 89k 76k 5, G(17 ;51; k) 361k 50k 18, G(9 ;41; k) 841k 8k, G(9 ;800; k) 841k 1600k 701 I the bove formule, whe k N, we c fd ftely my D such tht y 1 s solvble Pge - 5

Nmely, (5k 14k ) x y 1, (5k 36k 13) x y 1, (169 k 140k 9) x y 1, (169 k 198k 58) x y 1, (89k 76k 5) x y 1, (361k 50k 18) x y 1, (841k 8k ) x y 1 re solvble for y turl umber k, (841k 1600k 701) x y 1, Whe x s postve odd teger cotg prme fctor p whch belogs to or x 1, by gog through the dom of k, we obt the uo of fte sequeces For every term the sequeces, the Pell equto y Y s 1 s solvble d ll solvble D re coted these sequeces Tht s why ths theorem s ecessry d suffcet codto for determg the solvblty of form y 1 It lso gves wy to costruct D qudrtc Theorem7 Assume tht D Y, the lest postve soluto to x Dy 1 s ( x0, y 0), the y 1s solvble pys d p D, p x0 1 Proof (proof by cotrdcto) If p Ys d p D, such tht p x0 1 Sce y 1s solvble, suppose ( x, y) s soluto to t, ote tht (( y ) 1) 4 y ( y ) 4 y D x y y 4 4 ( y ) 1 The(( y ) 1, x y ) s soluto to x Dy 1 By lemm 3, for the solutos of y 1, we hve x x 1x0 x x1 x0 1 obt Becuse Deote pys d p D, such tht p x0 1, ccordg to our recurso formul, we c 1 x 1 (mod p), y by x m, the xm 1 (mod p), whch mes tht p x 1 m 16 / 4 Pge - 53

xm 1 Becuse y 0 (mod p ), we obt y 0 (mod p) However, y 1 1 (mod p), ledg to cotrdcto Coversely, If pys d p D, p x0 1The, by theorem, x0, y0, so x0 1 x0 1 y0 D We obt x0 1 p x0 1 x0 1 Becuse (, ) 1, by lemm 5, there re two tegers uv, such tht( uv, ) 1, d x0 1 Du, x0 1 v, Therefore, Du It mes tht ( uv, ) s soluto to By theorem 1, f x 1 x 1 v 1 0 0 y 1 y 1 s solvble, the DY or D U whe D U, the solvblty of y 1 s stll ukow Theorem7 gves suffcet d ecessry codto for y 1 to be solvble whe D Y However, ths theorem s ot sutble for determg the solvblty, f solvble, ssume tht the lest postve soluto to postve soluto to x Dy 1s ( x0, y 0) They stsfy the equltes x x, y y 0 0 0 0 y 1s y 1 s ( x 0, y 0) d the lest (We wll gve further llustrto the proof of theorem 8) If we use ths theorem for determg the solvblty, we hve to fd the lest postve soluto to x Dy 1 However, s the lest postve soluto to x Dy 1s greter th the lest postve soluto to y 1, tme complexty of solvg x Dy 1s greter th solvg y 1drectly Despte ths, the theorem ctully serves s rdge betwee the two problems here, 17 / 4 Pge - 54

mely the solvblty of y 1d the determto of soluto of x Dy 1 It s lso prctcl method f the lest postve soluto ( x0, y0) to x Dy 1s obvous sce oe merely eeds to verfy whether p x0 1to determe whether or ot the egtve Pell equto s solvble Theorem 8Assume tht D Y, d y 1 s solvble, by lemm 3 ts geerl soluto ( x, y ) ce gve by ts lest postve soluto ( x 0, y 0) through y y 1 ( 0 0), Let N D be the set of these solutos ( x, y ) Smlrly, by lemm, let P D be the set of geerl evely dexed solutos ( x, y) to x Dy 1 We c obt oe-to-oe mppg g : PD NDNmely, x y x 1, D y 1, Proof Assume tht D Y, the lest postve soluto to (314) x 0 Dy0 1 18 / 4 x Dy 1 s ( x0, y 0), the Assume tht p s prme fctor of D Tkg (314) modulo p, we obt (315) x0 1 (mod p) (315) s equvlet to x0 1 (mod p) If x0 1 (mod p), pply the recurrece relto of the soluto to lemm, we obt (316) x 1 (mod p), If y 1 s solvble, ssume tht ( x, y) s soluto Note tht (( y ) 1) 4 y ( y ) 4 y D x y y 4 4 ( y ) 1 The(( y ) 1, x y ) s soluto to x Dy 1 1 Deote y by x m, the by (316), xm 1 (mod p) Therefore, ( y ) 1 1(mod p) Hece, y 0 (mod p) x Dy 1by Pge - 55

Tkg y 1 modulo p, we obt 1 y 0 x 0 0 (mod p) Cotrdcto Therefore, for y prme fctor p of D, we hve, x0 1 (mod p) By lemm, we obt x 1 (mod p), x 1 (mod p), 1 Dy 1, we hve x, by theorem, we hve x, y Therefore x 1 x 1 y D Sce x 1(mod p), x, ( p,) 1, we obt x 1 p holds for y prme fctor p of D Therefore, by lemm 5, Implyg u, v Nmely, ( u, v) s soluto to, such tht x 1 Du, x 1 v, Du y 1 v 1 Here, x 1 u, D y 1 v, We obt mppg g from P D to N D, mely, x 1 x 1 g : ( x, y) (, ) D Note tht, j, by lemm, we obt x x, y y j j Therefore x 1 1 1 1 x j x x j, D D Ths shows tht g s jecto, ssume tht ( x, y) s soluto to y 1Note tht 19 / 4 Pge - 56

Therefore s soluto to x Dy 1 Sce (y 1) D( xy) ( y ) 4 y D x y y 4 4 ( y ) 1 (( y ) 1, x y ) ( y ) 11 y 1 u 1 y x, y, D D D holds for every ( x, y ) belogg to N D, there s verse mge belogg to P D Therefore, g s surjecto To coclude, g s oe-to-oe mppg 4 Relted work d expermets [3][4][5] proved tht the followg egtve Pell equtos re solvble: (41) x 5(5 ) y 1( 1(mod 4)), where 5 s prme umber; (4) x 5py 1, where p s Fermt prme d p 3,5 ; (43) x p( p ) y 1, where 1(mod 4) s prme, p 3(mod8) s prme Actully, the solvblty of these three egtve Pell equtos re strghtforwrd from theorem 3, becuse the gve D the three ppers follows from theorem 3 The proofs re show below For 5(5 ) 1( 1(mod 4)), where 5 s prme umber, x y sce 1(mod 4), we obt5 1(mod4) Sce 5 s prme umber, 5 belogs toy s Becuse 5 belogs toy s, substtute p to 5, q to 5, we obt p 5 5 1 q 5 5 5 5 Applyg theorem 3(), the bove egtve Pell equto (41) s solvble For x 5py 1, where p s Fermt prme d p 3,5 Deote 0 / 4 1 by p Becuse p 3,5, Therefore, p 1 1(mod 4), mely, p belogs toy s Also, Hece, 1 1 p 1 4 1 ( 1) 1 (mod 5) p 1 1 q 5 5 Applyg theorem3(), the bove egtve Pell equto(4)s solvble Pge - 57

For x p( p ) y 1, where 1(mod 4) s prme, p 3(mod8) s prme Deote p by q, becuse 1(mod 4), p 3(mod8), we obt q p ( 3)( 1) 1(mod 4) Implyg q belogs toy s Therefore, q p p p p p Sce p 3(mod8), we obt q 1 p p Applyg theorem 3(v), the bove egtve Pell equto (43) s solvble [9] dscussed the solvblty of x (4 ) y 1 Actully, the theorems the pper re strghtforwrd from Theorem 1, Theorem 3 etc For x (4 ) y 1 pper [9], the theorem 1 of the pper showed tht whe s odd umber d 1s prme, the bove egtve Pell equto s usolvble Here, D(1) d 1 3(mod4) Applyg theorem1, the bove egtve Pell equto s usolvble I [1], the theorem reveled tht D s solvble f d oly f t s the sum of two squres However, pplyg Fermt s theorem o sums of two squres, we obt tht the bove codto s equvlet to, DY or D U Whe usg the cotued frcto expso to clculte the prty of the perod, we cot reduce the power of the prme fctors of D to smller oe, becuse t my chge the prty of ts cotued frcto expso, whch wll ffect the solvblty However, our method c reduce the power of ech prme fctor to 1 or Ths mes tht the result ce geerlzed The prme umbers whch cogruet to 1 modulo4 d smller th 100 re lsted below 5,13,17,9,37,41,53,61,73,89,97 The vlues of Legedre symbols for ll possble prg of these umbers re lsted below p p q q 5 13 17 9 37 41 53 61 73 89 97 5 0 13-1 0 17-1 1 0 9 1 1-1 0 37-1 -1-1 -1 0 41 1-1 -1-1 1 0 53-1 1 1 1 1-1 0 61 1 1-1 -1-1 1-1 0 73-1 -1-1 -1 1 1-1 1 0 89 1-1 1-1 -1-1 1-1 1 0 97-1 -1-1 -1-1 -1 1 1 1 1 0 Tble the vlues of Legedre symbols Here re ll the D s by () of theorem 3 whch hve two dstct prme fctors d both prme fctor cogruet to 1 modulo 4(d smller th 100) There s t lest oe odd postve teger d b 1 / 4 Pge - 58

5 13 5 17 5 37 5 53 5 73 5 97 13 37 13 41 13 73 13 89 13 97 17 9 17 37 17 41 17 61 17 73 17 97 9 37 9 41 9 61 9 73 9 89 9 97 37 61 37 89 37 97 41 53 41 89 41 97 53 61 53 73 53 97 61 89 Tble 3 ll the Ds by () of theorem 3 whch hve two dstct prme fctors d both prme fctor cogruet to 1 modulo 4(d smller th 100) Here re ll the Ds by (v) of theorem3 whch hve two dstct prme fctors where oe of the fctor s d other prme umber cogruet to 1 modulo 4 (d smller th 50) Here, s postve teger 5 13 9 37 53 61 101 109 149 157 173 181 197 9 Tble 4 ll the Ds by (v) of theorem3 whch hve two dstct prme fctors where oe of the fctor s d other prme umber cogruet to 1 modulo 4 (d smller th 50) Here re some Ds by theorem 4 wth more th two dstct prme fctors where the prme fctors of these Ds cogruet to 1 modulo 4 Here, s odd postve teger c c 5 13 17 5 13 37 5 13 53 c c c 5 13 73 5 13 97 5 17 37 c c c 5 17 53 5 17 73 5 17 97 c c c 13 37 41 13 37 73 13 37 89 c c d c d 5 13 17 37 5 13 17 53 5 13 17 73 c d c d e c d e 5 13 17 37 53 5 13 17 37 73 5 13 17 37 97 c d e c d e f c d e f 5 13 17 37 53 73 5 13 17 37 53 97 5 13 17 37 73 97 c d e f g 5 13 17 37 53 73 97 c d e f g 17 9 37 41 61 73 89 c d e f c d e f g f 17 9 37 41 61 73 89 97 Tble 5 Ds by theorem 4 wth more th two dstct prme fctors where the prme fctors of these Ds cogruet to 1 modulo 4 All the umbers the bove 4 tbles re D s for whch y 1s solvble / 4 Pge - 59

5 Cocluso I secto 31, by observg the resdue of D, we gve ecessry codto for y 1to be solvble Therefore, secto 3 we m to fd the prme fctorzto of D Probg deeper, we cotued our reserch d proved theorem 3 d theorem 5 I the procedure, we used lemm 5 to sort out the fte cses, d the we used the propertes of qudrtc resdue to elmte ll the cses except two cses Moreover, we used the mmlty of the solutos ( x0, y0) to the postve Pell equto to elmte oe of two cses remg Flly, we costruct the soluto to the egtve Pell equto the lst remg cse I secto 33, we frstly show tht for gve soluto ( xy, ) to egtve Pell equto, we c fd D for whch y 1s solvble Therefore, by gog through ll solutos ( xy, ) to y 1, we c obt every D tht for whch y 1s solvble Frst, wth the property of qudrtc resdue, we obt x s odd umber wthout the prme fctors cogruet to 3 modulo 4; secodly, whe x 1, we obt Dk 1; thrdly, we c use the qudrtc cogruece equto to fd the lest postve teger y for every gve x such tht ( xy, ) stsfy y 1; flly, we determe ll possble D for whch y solvble Whe D Y, we hve the lest postve soluto ( x0, y0) to x Dy 1, mely (51) x 0 Dy0 1 Assumg p s prme fctor of D, tkg (51) modulo p, gves 1s x0 1 (mod p) Nmely, p x0 1 or p x0 1 By lemm, we obt resdue property for every soluto to x Dy 1 Ths leds to cotrdcto whe p x0 1 Therefore, we obt Theorem 7 I theorem 8, we buld oe-to-oe mppg from subset of the set of solutos of postve Pell equto to the set of solutos of egtve Pell equto 6 Future work Whe D U, the solvblty of y 1s ot completely work out However, we ve got results Such s, theorem3 dcte tht, whe D U, f D p, py,, the D p, py, s solvble s O the bss of these results, we hope to fd cocse theorem to determe the solvblty of y 1, whch s suffcet d ecessry codto for y 1to be solvble Becuse the solvblty of y 1ce used to determe the prty of the cotued frcto expso cycle of D type, we pl to fd out method to determe the prty of the perod of the cotued frcto expso for the postve teger D s tmes squre root Also, we pl to wrte progrm of WCJ lgorthm to determe the solvblty of egtve Pell equto utomtclly s 3 / 4 Pge - 60

Ackowledgemet We thks toour tutor MrJ for the fully support d help He s bee ccomped wth us durg ths strugglg tme ll log Thks to Professor Yg Jg d Mr Irvgfor ther dvce to us completgthepper Thks to Mr Qu for hs dvce Thks to our fred Lu Zyu for hs code Most mporttly, thks to C Lde s prets d Wg Jq s prets for ther support They could std for ther chldre ll dy log Referece [1] A Grytczuk, F Luc, M W ójtowcz, The egtve Pell equto d Pythgore trples, Proc Jp Acd, Volume 76 (000) 91 94 [] CP, CP, elemetry umber theory(bejg, Pekg Uversty Publc)003 ( Chese) [3] X Du, bout Pell equto x 5(5 ) y 1( 1 (mod4)) Jourl of hu uversty for toltes(nturl Scece edto)ju 01 179-181 [4] X Du, bout Pell equto x p( p ) y 1( p 3 (mod8)),jourl of southwest uversty for toltes(nturl Scece edto),01 181-18 [5] X Du, Determto o solutos uversty,01 55-57 4 / 4 x (4 ) y 1,Jourl of Sheyg [6] D Pumplu, Über de Klssezhl ud de Grudehet des reelqudrtsche Zhlkorper, J Ree Agew Mth 30 (1968), 167-10 [7] Edwrd Everett Whtford, the Pell equto, 191, 71 [8] F To, Sur quelques theorems de Drchlet, J Ree Agew Mth 105 (1889), 160-169 [9] X Gu, bout Pell equto x 5py 1,Jourl of southwest uversty for toltes(nturl Scece edto)010 3-33 [10] H W Lestr Jr Solvg the Pell Equto, NOTICES OF THE AMS VOLUME 49, NUMBER 00 18-19 [11] JC Lgrs, O the computtol complexty of determg the solvblty or usolvblty of the equto y 1, Trs Amer Mth Soc 60 (1980), 485 508 [1] K Hrdy, KSWllms, o the solvblty of the Dophte equto dv evw dw 1, PACIFIC JOURNAL OF MATHEMATICS Vol 14, No 1,1986 145-157 [13] P G L Drchlet, Ege eue S ätze über ubestmmte Glechuge Gesmmelte Werke, Chelse, New York, 19 36 [14] P Kpl, Sur le -groupe des clsses d'déux des corps qudrtques, J Ree Agew Mth 83/84 (1976), 313-363 [15] Testg the solublty of the egtve Pell equto, http://wwwumbertheoryorg/php/hrdy_wllmshtml, 013 Pge - 61