Integral points on hyperbolas over Z: A special case

Transcription

1 Itgral pots o hprbolas ovr Z: A spcal cas `Pag of 7

2 Kostat Zlator Dpartmt of Mathmatcs ad Computr Scc Rhod Islad Collg 600 Mout Plasat Avu Provdc, R.I , U.S.A. -mal addrss: ) Kzlator@rc.du ) Kostat_zlator@ahoo.com Effctv August, 009, ad for th acadmc ar 009-0; Kostat Zlator Dpartmt of Mathmatcs 30 Thackra Hall 39 Uvrst Plac Uvrst of Pttsburgh Pttsburgh, PA 560 U.S.A -mal addrss : kzt59@ptt.du `Pag of 7

3 . Itroducto Th subct mattr of ths work s tgral pot o cocs dscrbd b th stadard gral form quato α x x γ δx ε J 0 () ad whr th coffcts α,, γ, δ, ε, J ar tgrs satsfg th codtos 4αγ k, wth α 0, γ 0, ad k a postv tgr It s a wll-kow fact f a coc dscrbd b (), wth th sx coffcts bg ral umbrs ad 4αγ > 0; th such a coc must b thr a hprbola or a par of two trsctg straght ls (th dgrat cas). Thr s a xtsv bod of ltratur o cocs, spag a fw hudrd ars; but wth most books o th subct havg b publshd th last 50 ars. For xampl, th radr ma rfr to [].A tgral pot o a coc s smpl a ordrd par ( x, ) satsfg () ad wth both xad bg tgrs. Not that bcaus of codto (), quato () wh cosdrd as a dophat quato th varabls xad, caot b a Pll quato; sc th cas of Pll quato, α, γ d, δ ε 0, ad J or -; whr d s a postv tgr whch s ot a prfct or tgr squar; ad so () could ot b satsfd. A hprbola whch s dscrbd b a Pll-quato of th form x d ( d a oprfct squar), has fact ftl tgral pots. A proof of ths fact, as wll as th mthod of fdg all th solutos, ca b foud a umbr of umbr thor books. As xampls, s [] or [3].O th othr had, a Pll quato of th form x d wll thr hav ftl ma tgr solutos or o solutos at all; dpdg o th prod th smpl cotud fracto xpaso of th rratoal umbr d (s rfrcs [] ad [3]). Aga, as mtod bfor, o of th hprbolas w stud ths papr corrspods to a Pll I k δ 4αJ αε δ, plas a k rol. Wh quato. As w wll s, th tgr ( ) ( ) I 0, ach of hprbolas dscrbd b (), ad udr th codtos (), has ftl ma tgral pots (cludg th cas of zro or o tgral pots). Ths ftl ma tgr pars ( x, ) ca b foud b a mthod of tchqu outld Sctos ad 3. Ths mthod uss ol straghtforward algbra ad a coupl of basc facts o quadratc tromals. I Scto 4 w offr a umrcal xampl. I Scto 5, w offr a coupl of obsrvatos ad rmarks o thr tgr I abov. I Scto 7 w tak a look at th spcal cas α k. Ad Scto 8, w cosdr th cas I 0. Wh I 0, a coc dscrbd b () ad (), ow bcoms a par or uo of two trsctg straght ls. A straght l s dscrbd b a quato of th form ax b c; ad our cas wth a, b, cbg tgrs. Such a l ca thr hav o or ftl ma tgral pots. Solvg such a lar dophat `Pag 3 of 7 ()

4 quato s stadard,wll kow matral whch ca b foud vr troductor umbr thor book. So th cas I 0, th curv qusto s th uo of two trsctg straght ls. l : a x b c l : a x b c. ; wth a, b, a, b, c, c Z Ths ar xactl four possblts: l cotas ftl ma tgral pots ad lo; or vc-vrsa; or ach of thm cotas ftl ma tgral pots; or thr of thm dos.. A solvg tchqu Equato () s quvalt to α x x γ δx ε J λ λ (3) whr λs a ratoal umbr to b dtrmd. Cosdr th lft-had sd of (3) as a quadratc tromal x. W wrt t stadard form: ( δ ) x ( γ ε λ) λ α x J (4) Th dscrmat D( ) of ths tromal x, dpds o. W hav D( ) ( δ ) α ( γ ε J λ) 4, ad b () w obta ( ) k ( αε δ ) ( δ 4αJ 4αλ ) D (5) Th da hr s prtt smpl: to choos a ratoal umbr λsuch that D( ) bcoms th squar of a lar polomal ad of th form k r, whr rs a ratoal umbr; so that w ma furthr factor th tromal o th lft-had sd of (4) as a product of two lar polomals xad wth tgr coffcts. W choos λsuch that αε δ δ 4αJ 4αλ (6) k Th from (5) w obta, αε δ D( ) k k Solvg for λ (6) producs, `Pag 4 of 7 (7)

5 ( δ 4αJ ) ( αε δ ) k λ (8) 4αk Th lft-had sd of quato (4) ca b factord, accordg to th fudamtal Thorm of Algbra, as whr r ( ) ( r )( x r ( ) ) λ x ) α ( (9) ( δ ) D( ) ( δ ) D( ) ;, r ( ) α α ar th two roots (whch of cours dpd o ) of th quadratc tromal x. B (9),(8),(7), ad (4); ad aftr som algbra (whch cluds multplg both sds of (9) b 4α k ) w arrv at th dsrd factorzato: [ αk k( k) δk αε δ ] [ αkx k( k) δk ( αε δ )] k ( δ 4αJ ) ( αε δ ) (0) Lt k ( δ 4αJ ) ( αε δ ) I (0a) Not that f o wr to start wth quato (0); ad multpl out th two factors of th lft- had sd; us () ad collct th aggrgat x,, x, x,, ad costat trms; ach of ths trms wll cota a commo factor quato (). 4α k ; aftr th cacllato of whch, o rturs to 3. Th cas I 0 I δ α αε δ, ad ( ) Wh k ( 4 J ) ( ) 0 x, s a tgr soluto of (), ad hc of (0) as wll (ad covrsl), th th two factors o th lft-had sd of quato (0); wll b ozro dvsors of I, whos product s I. Lt d < d < K < d N I b th N postv tgrs whch ar th postv dvsors of th atural umbr I.I ordr to fd all th tgr pars ( x, ) satsfg (0), w must solv N lar sstms th ukows or varabls xad. W group ths N sstms to N groups, ach group cotag two sstms: Th th group αkx k αkx k ( k) δk αε d, ( ) ( ) k δk αε I d `Pag 5 of 7

6 Whr or (whch xplas wh ach group cotas two sstms); or quvaltl, stadard form, αkx k αkx k ( k) d δk ( αε δ ) I d ( ) ( ) k δk αε δ () W rmark hr that th tgr I has xactl N tgr dvsors: d, d,, d K N I ; ad thr gatv coutrparts d, d, K, d N. Wh o maks a choc for o of th two factors o th lft-had sd of (0); o chooss that factor I to b, whr or. Th th othr factor must qual ; so that th product of d th two factors quals I. Blow w us th wll-kow Kramr s rul from lar algbra ordr to solv ach lar sstm (). Frst, w comput th dtrmat of th matrx of th coffcts; whch must b ozro ordr to appl Kramr s. W hav, d αk dt αk k k ( k) ( k) αk 4αk 3 ( k( k) ) αk ( k( k) ) 0, scα 0 ad k s a postv tgr. Thus, ach lar sstm () wll hav a uqu soluto. Sc all th coffcts (cludg th costat trms) ar tgrs, t follows ach of th N sstm wll hav a uqu soluto ( x, ) whch both xad ar ratoal umbrs. Lt ( x, ) b thr of th two solutos (o for ach sstm) of th two sstms (); o sstm s obtad for, th othr for. If w hav to spcf whch o of th two solutos w ar rfrrg to; th x, wll stad for th soluto of th sstm () wth x, wll b th ( ) ; whl ( ) soluto of th sstm () wth. To b abl to wrt dow xplct formulas for xad ; w must comput two mor dtrmats as rqurd b Kramr s rul. Ths s stadard matral that ca b foud ot ol lar algbra txts, but also collg algbra ad prcalculus txts. Hr s th d rsult aftr som smplfg: d x I d ( k) ( k) δk ( αε δ ) d k I d 4αk ( αε δ ) () `Pag 6 of 7

7 4. A umrcal xampl Cosdr th hprbola wth quato x 5x x 0 (3) W hav α, 5, γ, δ, ε, J. Ad 4αγ 5 4()() 9 k ; k 3 Furthrmor w obta, αk, k k 4, δk 3, αε δ, k ( ) ( k) 6, δ 4αJ 9; ad thus b (0a), w also gt I 80. Th tgr 80 has xactl N 0postv dvsors; ad so th umbr of lar of sstms to b solvd s N 0. Howvr, f w look at quato (0); th cas of ths xampl w hav, ( 4 4)( x 6 ) 80 x (4) whch shows that w ca cacl out a commo factor from ach of th two sds of quato (3); ad stll hav two factors o th lft-had sd of th rsultg quato, ach of whch s a lar polomal xad wth tgr coffcts. Spcfcall quato (4) s quvalt to ( 3 6 )( 6x 3 ) 0 x (5) So ths obsrvato about th commo factor of 8 smpl xpdts th solvg procss. Equato (5) lds ght lar sstms: 3x 6 6x 3 0 3x 6 6x 3 5 3x 6 0 6x 3 3x 6 5 6x 3 3x 6 6x 3 0 3x 6 6x 3 5 3x 6 0 6x 3 3x 6 5 6x 3 ( a) ( b) ( c) ( d ) ( ) ( f ) ( g) ( h) `Pag 7 of 7

8 Of ths 8 sstms ol (b),(c),(f), ad (g) hav tgr solutos. Ths ar also th solutos of quato (3): (, ) (, ), (,0), (, ), ( 0, ) x. If w had procdd wthout cacllg th factorg th commo factor 8 quato (4); w would hav foud out that som of th rsultg 0 sstms ar fact quvalt; th d, of cours, w would hav foud th sam four tgr solutos abov. Also ot that th 0 postv tgr dvsors of 80 ar: d, d, d3 4, d 4 5, d5 8, d6 0, d7 6, d8 0, d9 40, ad d Obsrvatos ad rmarks Kp md that amog th N lar sstms o must solv ordr to fd all th tgr solutos of quato (0) ad thus of ()); thr ma a fw or svral groups, wth ach such group cotag quvalt sstms. Wh that happs, th whol solvg procss s sgfcatl smplfd/rducd. Now, lt us tak aothr look at th tgr ( δ ) 4α ( αε δ ) I k J. If or δ s v; th b spcto w s that I 0( mod 4). Ths statmt s obvous wh δ s v. Wh s v, th b 4αγ k, t follows that ks v as wll; so must I. If both ad δ ar odd tgrs; th so s k. Ad so, k δ ( αε δ ) ( mod 4) whch t asl follows that I 0( mod 4) rcall that th squar of a odd ad thus partcular to mod 4 Cocluso: Th tgr I s alwas a multpl of 4 6. Th cas δ ε 0., from tgr, s fact cogrut to( mod 8 ); ( ) Wh δ ε 0, quato () bcoms α x γ J (6) Accordg to (), w also hav 4αγ k (6a) `Pag 8 of 7

9 Equato (6a)sas that th product αγ must qual mus a tgr or prfct squar. That ca of cours occur wthout o of α ad γ bg a tgr squar whl th othr bg mus a prfct squar. For xamplα 8 ad 50 αγ γ, gvs ( ). Howvr, f o of α ad γ s a tgr squar whl th othr s mus a squar; th ths would b a suffct codto whl mpls that αγ s mus a tgr squar. Assum th that, α l ad γ m (6b) Whr lad mar postv tgrs. Ad so, th valu of th atural umbr ks k ml It s clar from (6) that f ( x, 0 0 ) s a tgr soluto, th so ar th pars ( x0, 0 ), ( x0, 0 ) ( x ). 0, 0 B (6) ad (6a) w gt, ad l x m J or quvaltl, ( lx m)( lx m) J (7) Not that f J s a tgr cogrut to modulo 4; th (7) has o tgr solutos. Ths s tru bcaus l x m ( lx) ( m) 0,, or 3( mod 4) accordg to whthr both lxad mar v or odd; lxs odd ad mv; or lxs v ad modd. Blow w xam th two cass wh J, p whr ps a odd prm; ad so J, p. Suppos J. Sc th ol postv dvsor of s. To fd all th tgr solutos of (7), w must solv th two lar sstms: lx m lx m lx m lx m ( ) ( ) l l J, quato (7) wll hav xactl two tgr solutos wh l,,0. Othrws, for a othr valu of th atural umbr l; ad for Th soluto of ( ) s ( x, ), 0 ad of ( ) s ( x, ), 0 W s that wh ths bg (,0) ad ( ) postv tgr m, quato (7) has o tgr solutos. `Pag 9 of 7

10 . Suppos J p. Th postv dvsors of par ad p. To fd all th tgr solutos of (7), w must solv th four lar sstms blow: ( ) ( v) ( v) ( v) lx m lx m p lx m p lx m lx m p lx m lx m lx m p Th solutos of ( ),( v), ( v), ad ( ) ( x ) v ar rspctvl, ( p ) p ( p ) p p p p p,,,,,, ad,. l m l m l m l m W coclud that wh (oxclusv) or m O th othr had, wh l J p, pad odd prm; ad l s ot a dvsor of ( p ) s a dvsor of ( p ) s ot a dvsor of ( ) p ; ; th quato (7) has o tgr solutos. ad m (7) has xactl four tgr solutos (whch ar lstd abov) 7. Th cas α k B () ad ( 0 a ) w hav s a dvsor ( p ), th quato 4γ ad I δ 4J ( ε δ ) (8) Not that must b odd ths cas, ad so ( mod8) ; ad thus γ v. Gog back to (), I d d x I d d wth or ( ) ( ) δ ( ε δ ) 4 ( ε δ ) (9) As w kow from th prvous scto, I s alwas a multpl of 4. Cosdr th spcal cas I,. `Pag 0 of 7

11 Th N postv dvsors of I ar, d, d, K, d, d d ; d, for, K, N I Not that wh ; d, d, ad thrfor s half a odd tgr; thus, s d I ot a tgr. Lkws wh,, but d d ; aga, s ot a tgr. O th othr had, for, both xad ar tgrs. Idd ths cas d ad I d odd, both ( ) ad ( ) ar both v, ad thrfor s v b spcto. Also, sc s ar v tgrs. Cosqutl, d ( ) ( ) 0( mod 4). Furthrmor, δ ( ε δ )( mod 4) I d, sc ( mod ) ; ad so ( mod 4). Wh δ s odd, th ( ε δ ) ( mod 4) whl wh δ s v δ ( ε δ ) 0( mod 4). W s that wh I, δ ;, ad th codtos (8) ar satsfd, th hprbola dscrbd b quato () cotas xactl I ( ) (dstct, s rmark blow) tgral pots. Usg d ad, th formulas (9), ad aftr som smplfg, w ca stat th followg thorm. d Thorm Suppos that α,, γ, δ, ε, J ar tgrs such that α, 4γ, γ 0, ad I δ 4J ( ε δ ),. Th, th hprbola dscrbd b quato () cotas prcsl ( ) tgral pots gv b th formulas ( ) ( ) δ ( ε δ ) x ε δ for, K, ; ad wth or Rmark. I th last scto, Scto 0, w prov that th ( ) tgral pots of Thorm, ar fact dstct. 8. Th cas I 0 `Pag of 7

12 Wh I k ( 4 J ) ( ) 0 δ α αε δ, th th curv dscrbd b () ad (), s a par of two trsctg straght ls. I ths cas, th soluto sts of (0) ad thrfor of () as wll s th uo of two sts: S S S, whr Ss th soluto st of th lar dophat quato, ( k) δk αε 0 αkx k δ ( k) δ ( αε δ ) ; or quvaltl αkx k k (0a) Ad Ss th soluto st of th lar dophat quato ( k) δ ( αε δ ) αkx k k (0b) All four possblts ca occur: S φ(mpt st) ad S φ ; S φ ad S φ ; S φ; or S S φ ad S φ. I both cass of (0a) ad (0b); w ar dalg wth a lar dophat quato two varabls: ax b c ; wth a, b, c bg tgrs. Ths s wll kow, stadard matral, that ca b foud almost vr troductor umbr thor book. Such a quato has thr o tgr solutos or ftl ma solutos. It has ftl ma tgr solutos f, ad ol f, th gratst commo dvsor gcd( a,b) of th coffcts aad b; s also a dvsor of c. Thr s a wll kow mthod for fdg all th tgr solutos of a two-varabl lar dophat quato wth tgr coffcts; a procdur basd o what s kow th ltratur as th Euclda algorthm for fdg th gratst commo dvsor of two tgrs. I th d, all th tgr solutos ca b paramtrcall xprssd trms of o tgr paramtr. Th radr ma rfr to [] or [3] for furthr dtals. I th cas of th lar dophat quato (0a); quato (0a)wll hav ftl ma tgr solutos f, ad ol f, gcd( α k, k( k) ) k gcd( α, k) of th tgr δ ( αε δ ) s a dvsor k. If that s th cas, th st Swll cota ftl ma tgral pots; othrws φ. Lkws, quato (0b) wll hav ftl tgr solutos xactl S wh gcd( α k, k( k) ) k gcd( α, k) s a dvsor of th tgr δk ( αε ) th cas, th st Swll cota ftl ma tgral pots. Othrws S φ.. If ths s 9. Th cas δ ε J 0 `Pag of 7

13 It mmdatl follows from (0a) that I 0. Also ot that ths cas, quato () s a homogous quadratc quato xad (.. vr moomal trm has dgr ). Sc I 0, quatos (0a)ad(0b) from th prvous scto, do appl. w obta ( k) αkx k 0 or quvaltl, αx ( k) 0 (a) Ad also (0b)bcoms ( k) αkx k 0 or quvaltl, αx ( k) 0 (b) I ths spcal cas, th coc dscrbd b quatos () ad (); s a par of two trsctg straght ls, wth thr pot of trscto bg th org ( 0,0). Blow w fd th sts Sad S. Frst S : w fd all th tgr solutos to (a) Lt d gcd( α, k). Th α d ρ ad k d v, whr v ad ρar rlatvl prm tgrs; gcd( v, ρ). Accordgl, b (a) w obta, ρ v (a) x Sc ρs rlatvl prm to v ; ad thus to vas wll; ad t dvds th product v ; t must b a dvsor of (ths s th wll kow Eucld s Lmma; whch s fudamtal provg th Fudamtal Thorm (Uqu Factorzato) of Arthmtc) W st Thrfor from (a) w obta x v t. Thus, {( x ) x v t, t t Z} S, ρ, ; whr ρad var th tgrs dfd b: ρ gcd α, ad v k ( α, k ) gcd( α, k) Nxt, f w lt d gcd( α, k) wth ( ρ, v ). gcd ρ t; t a tgr. ad w st α d ρ ad k d v ; ρ, v Z (b) W obta from th quato `Pag 3 of 7

14 `Pag 4 of 7 v x ρ (b) Workg smlarl as wth th prvous cas, w fd that ( ) { } Z t t t v x x S,,, ρ whr ( ) ( )., gcd,, gcd k k v k α α α ρ 0. Th dstctss of th ( ) tgral pots Thorm I ths scto w prov that th ( ) tgral pots, as dfd Thorm, ar dstct. W hav, ( ) ( ) ( ) δ ε δ ε δ x (3) Rcall that 4 γ, ad so s odd. Frst w show that for ach ( ),, K, th two pars ( ) x, ar dstct. O of ths two pars s obtad wh ; w wll dot t b ( ) x, ; th othr corrspods to, w wll dot t b ( ) x,. Lt us compar thr -coordats. Wh ca t b? W hav, δ ε δ ε whch s possbl ol wh s v. But th, for, ca th x-coordats b qual? W hav ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). x x 0, a obvous mpossblt.

15 Nxt, w prov that o two tgral pots ( ) th that <. I partcular. For th pot ( x, ) w wll us ( x, ), w wll us Frst, lt us xam th possblt x, ad ( ) x,, ca cocd wh. Assum th formulas (3); whr or. Lkws for th pot th formulas (3); whr or.. Ths s quvalt to [ ] [ ] (4) ; both tgrs ( ) ad ( ) Sc ar odd; thrfor quato(4) mpls th two powrs of o thr sd of (4) must b qual; whch mas that w must hav ; (5) Accordgl (4) lds, ( )( ) 0 (6) Howvr ; ad so > 0. Thus (6) mpls 0; whch mas that thr ad or ad (7) W hav s that, prcsl wh th codtos (5) ad (7) ar satsfd. Now, ca addto to ths, also hav x x? Gog back to (3), w s that x x s quvalt to ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] `Pag 5 of 7

16 `Pag 6 of 7 From (5) w kow that. Substtutg for th last quato lds ( )[ ] ( )[ ] Also, b (7), w hav Thus, th last quato mpls ( ) [ ] ( ) [ ] ; whch gvs, a mpossblt. Ths provs that ( ) ( ) x x, for all ad such that <. W ar do

17 Rfrcs [] L.P. Eshart, Coordat Gomtr, 98 pags. Frst publshd b G ad Compa 939. Rpublshd b Dovr Publcatos, Nw York, NY (960) Thrd Edto b Dovr Publcatos, Nw York, NY (005) ISBN-3 : , ISBN: , s p.p 08-8 [] Kth H. Ros, Elmtar Numbr Thor ad Its Applcatos, Ffth Edto, Parso Addso Wsl, 7 pags, 005. For Pll s Equato: s p.p For Cotud Fractos: s p.p For Eucld s Lmma: s pag 09 For Lar Dophat Equatos: s pags [3] W. Srpsk, Elmtar Thor of Numbrs, Warsaw, Polad, 964, 480 pags Rprtd 988 b North Hollad. ISBN: For Pll s Equato : s pags Also pags For Cotud Fractos of quadratc rratoals: s pags For Eucld s Lmma: s pag 4 ( Thorm 5) For Lar dophat quatos: s pags 7-3 `Pag 7 of 7