Modified Farey Trees and Pythagorean Triples
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1 Modified Frey Trees d Pythgore Triples By Shi-ihi Kty Deprtet of Mthetil Siees, Fulty of Itegrted Arts d Siees, The Uiversity of Tokushi, Tokushi 0-0, JAPAN e-il ddress : kty@istokushi-ujp Abstrt I 6, F J M Brig disovered terry tree of priitive Pythgore triples, where eh triple is trsfored to other three triples by three distit uiodulr tries This ft hs bee redisovered y ties I this pper, we shll give eleetry expltio of this ft usig lssil Eulide pretriztio of priitive Pythgore triples d odified terry Frey trees 00 Mthetis Subjet Clssifitio PriryD0; Seodry B Itrodutio I his pper [], Brig foud the followig iterestig pretriztio of the priitive Pythgore triple The priitive Pythgore triple is the set of positive itegers (, b, whih stisfy + b, with (, b Fro the oditio (, b, d b ust stisfy b od Therefore, without loss of geerlity, we y ssue y priitive Pythgore triple (, b, with odd d b eve i the followig Brig gve the followig uiodulr tries M, M, M,
2 where Brig s origil M is the bove M d Brig s M is the bove M Propositio (Brig [] Ay priitive Pythgore triple (, b, hs the uique represettio s the trix produt M σ( M σ( M σ(r b for soe r 0, (σ(, σ(,, σ(r (,, r, It is well kow tht Eulid hs desribed pretriztio of priitive Pythgore triples i his book Eleets s follows Propositio (Theore of [] Ay priitive Pythgore triple + b, with b be uiquely represeted by, b, +, with (, d > > 0 Moreover d ust stisfy the oditio od Fro these propositios, there exists the followig well kow bijetio Redued frtios with > > 0 d od Priitive Pythgore triple (, b, with b Modified Frey trees The Frey series of order N, deoted by F N is the set of ll redued frtios betwee 0 d whose deoitors re N or less, d rrges i iresig order For exple, if N, we hve F { 0,,,,,,,,,, } The followig tree is the usul Frey tree osistig of Frey series F N (N
3 Skippig the redued frtios with, odd i the bove Frey tree, we shll obti the followig odified Frey tree of redued frtios, where, re of odd prity d > > 0 6 Here the brh betwee d i this tree es the skipped frtio, other brhes lso represet the skipped frtios with od Now, we shll ll the frtio the frtio of level d the frtios,, the frtios of level d so o Thus, for y, there exist frtios of level, frtios of level + d brhes whih orrespod to the skipped frtios betwee level d level + For eh, we shll reple the frtios of level eh other so s the brhes to be pled t the left hd side of the lies betwee the frtios of level d level For exple, i the level, is hged ple with i the followig tree Here the brh whih orrespods to the skipped frtio is pled t the left hd side of the lie betwee the frtio of level of level d the frtio
4 (/ terry Frey tree 6 We will ll this odified Frey tree by (/ terry Frey tree The eh frtio i the bove (/ terry Frey tree of level orrespods to brh betwee the frtios of level d the frtios of level +, tht is, skipped frtio with, odd bijetively Hee we ostrut other odified Frey tree fro this (/ terry Frey tree s follows We ote tht we hve to trspose the redued frtios syetrilly with respet to the eter lie through the frtio so s to oute Brig s Pythgore tree (/ terry Frey tree We shll ll this odified Frey tree by (/ terry Frey tree Uiodulr trix tree I this setio, we shll relte eh redued frtio i (/ terry Frey tree to uiodulr trix By virtue of the exteded Eulide lgorith, we fid uique positive itegers x <, y < whih stisfy the followig lier diophtie equtio for y redued frtio ( > > 0, x y
5 Put x x, y y The y x,, y re suessive Frey series of order Sie d re of odd prity, ext oe of x, x, y, y is eve We deote by d oe of x x d y y whih stisfies d od We deote other reiig frtio by b The we kow the followig lier frtiol trsfortio ( b d x ( Thus the redued frtio ( b d orrespods to the trix oe to oe Hee we hve verified tht there exists the followig bijetio; Redued frtios with > > 0 d od ( b d Uiodulr tries More preisely, we hve bijetio fro the followig essetil prt of (/ terry Frey tree + b + + b + + b + + d + to the followig orrespodig prt of the tree of uiodulr tries; ( b d ( ( ( b b + d b + d b + d b + d d Let F be the trix suh tht ( b d ( b + d d F +
6 ( 0 The we hve F stisfy ( b d ( b + d b + d F + + Siilrly let F, F be the tries whih ( b d, ( b b + d F + ( ( The we hve F d F Now we hve obtied the 0 followig uiodulr trix tree fro (/ terry Frey tree Sie we shll ostrut other uiodulr trix tree fro (/ terry Frey tree, we will ll the followig uiodulr trix tree (/ uiodulr trix tree (/ uiodulr trix tree ( 0 ( 0 ( ( ( 0 ( ( ( ( ( ( ( ( Hee the trix A i the bove tree orrespods : to the redued frtio i (/ terry Frey tree by the lier frtiol trsfortio ( A Sie ( ( 0, we get the followig propositio idutively Propositio For y r 0, y redued frtio of level r+, where > > 0 d od hs the uique lier frtiol trsfortio represeted s the trix produt ( 0, where (σ(, σ(,, σ(r (,, r F σ( F σ( F σ(r ( 6
7 Let A i be the uiodulr trix whih stisfies A i ( 0 The oe kows tht ( 0 A Now we hve ( 0 ( ( 0 A σ( A σ(r ( F i ( 0 ( 0, A ( F σ( F σ( F σ(r ( 0 F σ(, for i ( ( 0 ( 0, A F σ(r ( 0 ( 0 ( Theore For y r 0, y redued frtio of level r +, where > > 0 d od hs the uique lier frtiol trsfortio represeted s the trix produt ( A σ(a σ( A σ(r, where (σ(, σ(,, σ(r (,, r Reltio to Brig s tries Fro the bove theore, we kow eh redued frtio i (/ terry tree is trsfored to other three redued frtios by the followig three distit tries A, A d A s follows ( ( 0 ( A ( ( 0 ( A ( ( 0 ( A, +, +
8 Fro the lssil Eulide pretriztio of priitive Pythgore triples, the redued frtio orrespods to the priitive Pythgore triple (,, d eh trsfortio A i ( i iplies trsfortio of the priitive Pythgore triple b + b + I the se A, we hve ( + ( (+( + b+, b ( ( (+( + b+, ( + + ( (+( + b+ Thus the trsfortio of the priitive Pythgore triple M idued fro A is defied by b Hee we hve obtied M Brig [] I the se A, we hve b + b + b + M b, whih ws deoted by M i (+ ++ ( +(+( + +b+, b (+ + ( +(+( + +b+, ( ( +(+( + +b+ Thus the trsfortio of the priitive Pythgore triple M idued fro A is defied by b Hee we hve obtied M Brig [] I the se A, we hve + b + + b + + b + M b, whih ws deoted by M i (+ ++ ( +(+( + +b+,
9 b (+ + ( +(+( + +b+, ( ( +(+( + +b+ Thus the trsfortio of the priitive Pythgore triple M idued fro A is defied by b Hee we hve obtied M + b + + b + + b + M b, whih is deoted by M i Brig [] Thus we hve give very eleetry expltio of the reso why Brig s three uiodulr tries geerte ll the priitive Pythgore triples The se of (/ terry tree I this setio, usig (/ terry Frey tree, we shll show other Eulide pretriztio idues the se represettio of Pytgore triples of Brig Usig the ottios i setio, we hve the followig essetil prt of (/ terry Frey tree d orrespodig uopdulr tries b + + b d + where +, b + d with d od ( b b + + ( b ( ( b + b b Now we will ll the followig uiodulr trix tree orrespodig to
10 (/ terry Frey tree, (/ uiodulr trix tree (/ uiodulr trix tree ( 0 ( ( ( ( 0 ( ( ( ( ( ( ( ( 0 6 Let G be the trix suh tht ( ( b b b + G + The we hve G stisfy ( b ( 0 ( + b G + Siilrly let G, G be the tries whih ( b, ( b G ( ( 0 0 Thus we hve G d G, respetively I the se wy s (/ terry Frey tree, we kow tht ( ( 0 d get the followig propositio idutively Propositio Ay redued frtio with > > 0 d od hs the uique lier frtil trsfortio represeted s the trix produt ( 0 ( G σ( G σ( G σ(r, for soe r 0, (σ(, σ(,, σ(r (,, r 0
11 Let B i be the uiodulr tries whih stisfy B i ( 0 The oe kows tht ( 0 B A, B Hee we hve ( 0 ( ( 0 G i ( 0 ( 0 ( G σ( G σ( G σ(r ( 0 G σ( A σ( A σ(r (, where i A, B ( ( 0 ( 0 G σ(r ( 0 A ( 0 ( Theore Ay redued frtio with > > 0 d od hs the uique lier frtiol trsfortio represeted s the trix produt A σ( A σ( A σ(r for soe r 0, (σ(, σ(,, σ(r (,, r (, Coludig rerks There exist two pretriztios of the Pythgore triple by (/ terry Frey tree d (/ terry Frey tree I the followig, we shll show these two pretriztios idue the se Brig s tree d the se uioduler tries The bijetio betwee the redued frtio i (/ terry Frey tree d the redued frtio i (/ terry Frey tree is give by the trix trfortio ( (, d oversely ( The we hve the followig outtive digr: ( (
12 A i A i ( A i A i ( Here we hve used the fts ( ( A i A i, for y i We lso hve the orrespodee of two Eulid s pretriztio s follows; + b + Now it is esily verified tht there exists the followig outtive digr of three represettios; M σ( M σ( M σ(r b A σ( A σ( A σ(r ( A σ( A σ( A σ(r ( Rerk Sie the bove expltios re very eleetry d strightforwrd, these results ust be lredy kow to the speilists But, to the best of y kowledge, I hve ever see y literture whih write dow these fts expliitly Thus it will be of soe worth for writig these fts expliitly i this ote Filly, we shll surize bove results i the followig digr
13 Bijetive reltios of trees d represettio Tree of priitive Pythgore triples (/ terry Frey tree (/ terry Frey tree Brig s uiodulr tries represettio (/ uiodulr trix tree (/ uiodulr trix tree Referees [ ] R C Alperi, The odulr tree of Pythgors, Aeri Mthetil Mothly,, (00, 0-6 [ ] F J M Brig, O Pythgore d qusi-pythgore trigles d geertio proess with the help of uiodulr tries (Duth, Mth Cetru Asterd Afd Zuivere Wisk, ZW-0(6 pp [ ] A Hll, Geelogy of Pythgore trids, Mth Gzette,, (0, - [ ] G H Hrdy d E M Wright, A Itrodutio to the Theory of Nubers, th ed Oxford Uiversity Press, Oxford, [ ] D Roik, The dyis of Pythgore triples, Trs Aer Mth So, 60, (00,
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