Torsion in Groups of Integral Triangles


 Ralph Herbert Hoover
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1 Advnces in Pure Mthemtics, 01,, Pulished Online Jnury 01 ( Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics, Cliforni Stte University, Long Bech, USA Emil: Received August, 01; revised Octoer 19, 01; ccepted Novemer 5, 01 ABSTRACT Let 0 π e fixed pythgoren ngle We study the elin group H of primitive integrl tringles c,, for which the ngle opposite side c is Addition in H is defined y dding the ngles opposite side nd modding out y π The only H for which the structure is known is H π, which is free elin We prove tht for 1 cos is rtionl squre, nd it hs elements of order three iff generl, H hs n element of order two iff the cuic x x 1 0 hs H hs twotorsion is dense in simil either or in cos r tionl solution 0 x 1 This shows tht the set of vlues of for whic h 0, π, nd rly for threetorsion We lso show tht there is t most one copy of H Finlly, we give some exmples of higher order torsion elements in H Keywords: Aelin Groups; Cuic Equtions; Exmples; Free Aelin; Geometric Constructions; Group Theory; Integrl Tringles; Lw of Cosines; Primitive; Pythgoren Angles; Pythgoren Tringles; Pythgoren Triples; Rtionl Squres, ThreeTorsion; Torsion; TorsionFree; TwoTorsion; Tringle Geometry 1 Introduction In [1], Olg Tussky introduces group opertion on the set of primitive pythgoren triples c,, :gcd c,, 1, c, tht is, the set of primitive integrl tringles c,, in which the ngle opposite side c is right ngle Ernest J Eckert spells out the detils of the group opertion in []: c,, ABC,, A B, B A, cc if A B 0, : B A, B A, cc if A B 0, where we reduce the nswer y common fctor if necessry to otin primitive triple The group hs identity 1, 0,1; the inverse of c,, is c,, The construction rises from the fmous identity on sums of squres: B B A B B A This identity, of course, corresponds to the rule z w zw for complex numers, nd the group lw on pythgoren triples hs similr geometric motivtion: we identify the tringle c,, with the complex numer i, nd the group opertion is then just complex multipliction The rgument of i is the ngle opposite side of the tringle Since we dd the rguments of complex numers when we multiply the numers, we cn interpret the group opertion s ddition of the ngles 1 nd to find primitive triple whose correspondπ ing ngle is 1 A prolem rises when 1, ie when A B 0, since this no longer corresponds to pythgoren triple with positive coordintes To fix this, we mod out the ngle y π ; this is why we hve the two cses in the definition of the group opertion ove This group hs een the suject of much interest over the yers In [], Jen Mrini studies the group of homogeneous liner trnsformtions with integrl coefficients tht preserve the tringles ove s set In [4], Brr Mrgolius uses sequence of pythgoren triples derived from the group lw ove to prove tht 1 rctn Plouffe s constnt is trnscendentl, specil π cse of Hilert s seventh prolem Eckert s min result in [] is tht this group is free elin, tht is, isomorphic to direct sum of copies of the integers The sum is indexed y the tringles,, p, where nd p is prime with p 1mo d4 In prticulr, the group is torsionfree, tht is, there do not Copyright 01 SciRes
2 exist ny nontrivil elements x 0 with nx 0 for n 0 In [5], Eckert nd Preen Dhl Vestergrd generlize the construction ove to the cse when is no longer right ngle We define n ngle 0 π to e pythgoren if or π is n ngle in pythgoren tringle, or, equivlently, if sin nd cos re oth rtionl For fixed pythgoren ngle, we define H to e the set of primitive integrl tringles,,c for which the ngle opposite side c is We cn dd two tringles y dding their vlues for, modulo π : c,, ABC,, A B, B A B cos, cc : B A Ac os, B A, cc We use the first formul ove in the cse when A B 0, which corresponds to 1 π, nd the second when A B 0, which corresponds to 1 π As efore, we stipulte tht we will scle the nswer y common fctor if necessry to otin primitive integrl tringle Eckert nd Vestergrd prove tht this construction gin gives us group, still with identity 1, 0,1 nd with the inverse of c,, eing c,, Note tht the originl group considered in [] is just the specil cse H π, whose structure we know completely Eckert nd Vestergrd sk numer of interesting questions out the structure of H in generl They π mention tht when, cn hve nontrivil torsion elements, tht is, elements x 0 such tht nx 0 for some n 0, ut they do not give detils In this pper, we pursue the study of torsion elements in H Our min results give necessry nd sufficient conditions on for H to hve twotorsion nd threetorsion From these theorems, it will e esy to see tht the set of vlues of for which H hs twotorsion is dense in the intervl 0 π, nd similrly for threetorsion We will lso show tht there is t most one copy of either or in H Finlly, we will give some exmples of higherorder torsion elements in H TwoTorsion Our results on twotorsion nd threetorsion re similr in flvor, ut the theorem on twotorsion is simpler nd significntly esier to prove, so we will strt there Theorem 1 H hs point of order two iff 1 cos is rtionl squre When this is true, the point of order two is unique Proof of Theorem 1 Since the inverse of c,, is c,,, point hs order two iff it hs the form c,, W MURRAY 117 Suppose first tht c,, H is point of order two We invoke the Lw of Cosines: So 1 cos c cos c 1 cos is rtionl squre, s desired c Conversely, suppose tht 1 cos, where, c re reltively prime Note tht 0 1cos 4, so 0c, so c,, is tringle, nd the computtion ove shows tht y the Lw of Cosines, the ngle opposite side c is Hence c,, H is point of order two To show tht only one point of order two cn exist, we note tht ny two tringles c,, nd,, c in H would hve to e sclr multiples of ech other, nd would therefore e represented y single primitive integrl tringle We illustrte Theorem 1 with some quick exmples: 1) For π, we hve 1 cos, nd s shown in [], hs no point of order two ) For π, we hve 1 cos 1, nd H π contins the point (1, 1, 1) of order two 1 ) For rccos 8, we hve 9 1cos, 4 nd H contins the point (,, ) of order two Corollry The set of for which H contins point of order two is dense in 0, π Proof This follows from the fct tht 1 cos is continuous function from 0, π onto 0, 4 nd the rtionl squres re dense in 0, 4 H ThreeTorsion Our result for threetorsion follows the sme generl spirit s tht for twotorsion, ut is significntly hrder We first prove the esy result tht H contins t most one copy of ; this proof will help set the stge for the proof of our min theorem Theorem H contins t most two points of order three Proof of Theorem : Suppose c,, H hs order three Note tht its inverse c,, lso hs order three, so we my ssume without loss of generlity tht (If, then s we sw ove,,, c hs order two, so it cnnot hve order three) Let nd e the ngles opposite sides nd respectively Since, we hve, so π Thus, when we compute c,,, we re dding to itself, nd there is no modding out y π The resulting tringle is,c, c,,, which hs ngle opposite side, so But since is fixed, the eqution Copyright 01 SciRes
3 118 W MURRAY completely determines the three ngles of the tringle, so there is only one primitive triple with these ngles (The other point of order three, of course, is the inverse c,, ) We will continue to use some of the ides ove in the proof of our min theorem on threetorsion Theorem 4 H hs point of order three iff the cuic cos x x cos 4 8 cos 4 cos Rerrnge: 4 4 cos 8 cos 4 cos 0 Fctor: cos cos 0 We clim tht cos 0 To prove this clim, hs rtionl solution 0 x 1 suppose tht cos 0; then cos Plug The cuic criterion gives esy nswers to two questions: First, given specified, we cn sk whether ging this into the second scling eqution, we get c =, so = c Plugging this into the first scling eqution, we H hs threetorsion Assuming tht is pythgoren, we cn use the Rtionl Root Theorem to check quickly get, contrdicting the fct tht nd re oth whether the cuic ove hs rtionl solution etween integers This proves our clim, so we my cncel the 0 nd 1 If it does, the proof elow will give constructive lgorithm to produce tringle c,, of order cos 0 fctor of cos : three Second, if we don t hve specific in mind, we Divide y : cn use the cuic criterion to find vlues of for which cos 1 0 H hs threetorsion Indeed, we cn solve the eqution ove for cos : x 1 cos x The rtionl function on the right tkes vlues less thn 1 for 0 < x < 1/, nd it mps the intervl 1/ x 1 continuously onto the rnge 1,1 Therefore, to find ngles whose groups hve threetorsion, we cn tke ny rtionl 1/ x 1 nd then find the corresponding Proof of Theorem 4 First, suppose tht c,, hs order three As in the proof of Theorem ove, we my ssume tht > nd tht when we compute,c,, there is no modding out y π So we use the first set of equtions for the group opertion: c,, cos, c ~ c,,, The common fctor etween the two tringles on the right is c, giving us the following scling equtions : c c cos cos Solve the second scling eqution for c: c 1 cos Plug this into the Lw of Cosines: cos 4 1 cos cos Multiply y : Set x :, which is rtionl numer with 0 < x < 1, nd we hve x x cos 1 0, s desired Conversely, suppose we hve rtionl solution 0 < x < 1 to x x cos 1 0, Define to e positive reltively prime integers such tht x, nd define c to stisfy c cos (Note tht c > 0) Scle,, nd c y common fctor, if necessry, to e reltively prime integers; note tht this scling will preserve oth x nd c cos By reversing the equtions from the forwrd direction of the proof, we cn work ck to cos 0 We cn solve this for cos nd then uild up to cos s follows: cos cos Copyright 01 SciRes
4 W MURRAY 119 Sustitute: Simplify: cos cos c c Now we check tht c,, is tringle First, we lredy sw tht, so c Second, to see tht c, we cn reorgnize the eqution immeditely ove: c c c Third, we cn continue to reverse the equtions from the forwrd direction of the proof until we get ck to cos 4 1 cos cos c, 7 ) Let rccos Then the cuic ecomes x x 0, which hs the rtionl solution 4 x This leds to the tringle 8,16,?H, which 7 hs order three Although it is not quite so ovious, we hve the sme corollry s for twotorsion: Corollry 5 The set of for which H contins point of order three is dense in 0, π Proof The rtionls re dense in the intervl 1/ x 1, which, s noted ove, is mpped continuously y x 1 onto 1,1 nd in turn y rccos onto 0 y x π Hence the set of corresponding to rtionl x is dense in 0, π 4 Exmples of Higher Order Torsion We close with some exmples of higher order torsion points in H : 1 1) Let rccos Then the tringle (10, 8, ) hs order four in H ) Let which gives us c It lso shows y the Lw of hs order five in H Cosines tht the tringle c,,, hs ngle opposite side c, so c,, H ) Let Finlly, we hve lredy estlished the scling equtions which show tht c c cos c c c c,,, hs order three, s desired,,, cos, ~,, so c,, Here re some smple pplictions of Theorem 4: 1) Let π Then the cuic degenertes to x 10, which hs no rtionl solutions t ll, nd H π hs no point of order three 9 ) Let rccos Then the cuic ecomes x x 0, which hs the rtionl solution x Following this through the proof ove leds to the tringle 6, 4,5 H, which hs order three 475 rccos Then the tringle (105, 81, 1) rccos Then the tringle (10, 104, 819 1) hs order six in H I found these essentilly y rute force serch on computer, finding multiples of integrl tringles nd seeing which ones hve finite order As ptterns strt to emerge for smller vlues, we cn nrrow nd speed up the serch for lrger vlues y restricting to those tringles tht stisfy similr ptterns In prticulr, there is 4 5 cler pttern in the vlues of here:,,4, From computergenerted evidence, the pttern on the exponents seems to e genuine, ut the pttern on the ses is red herring: other exmples of points of order six re (5555, 15, 04) for rccos, (1714, 7776, ) for rccos, nd (48, 16807, 40) for rccos The consistent pttern seems to n1 e tht point of order n must hve the form k,, c n (or, of course, k 1,, c ), ut I do not hve proof of this I invite you to prove this or find counterexmple Copyright 01 SciRes
5 10 W MURRAY 5 Conclusions nd Further Questions We hve seen tht lthough H π is torsionfree, there re mny vlues 0, π for which H hs torsion points of vrious orders In fct, the set of for which H hs twotorsion (respectively, threetorsion) is dense in 0, π, nd we cn esily chrcterize ll such in terms of conditions on cos We hve lso seen tht in such cses, H contins unique copy of (respectively, ) There remin mny intriguing open questions on H Oviously we would like to hve complete determintion of the isomorphism type of H for ll, ut since tht doesn t seem fesile in the ner future, there re plenty of more pprochle issues: 1) For wht vlues of n do there exist H with ntorsion? ) For ny n for which this is true, how cn we chrcterize the for which H hs ntorsion? ) Is the set of such dense in 0, π? (Note tht in ll the exmples of higher order torsion ove, we hd π ) 4) Is it possile for H to contin more thn one copy of n? REFERENCES [1] O Tussky, Sums of Squres, Americn Mthemticl Monthly, Vol 77, No 8, 1970, pp doi:1007/17016 [] E J Eckert, The Group of Primitive Pythgoren Tringles, Mthemtics Mgzine, Vol 57, No 1, 1984, pp 7 doi:1007/69091 [] J Mrini, The Group of the Pythgoren Numers, Americn Mthemticl Monthly, Vol 69, 196, pp doi:1007/1540 [4] B H Mrgolius, Plouffe s Constnt is Trnscendentl [5] E J Eckert nd P D Vestergrd, Groups of Integrl Tringles, Fioncci Qurterly, Vol 7, No 5, 1989, pp Copyright 01 SciRes
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