A BASIS OF THE GROUP OF PRIMITIVE ALMOST PYTHAGOREAN TRIPLES

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1 Joural of Algebra Number Theory: Advaces ad Applcatos Volume 6 Number 6 Pages 5-7 Avalable at DOI: A BASIS OF THE GROUP OF PRIMITIVE ALMOST PYTHAGOREAN TRIPLES Departmet of Mathematcs Sea College 55 Loudo Road Loudovlle NY USA e-mal: rylov@sea.edu Abstract Let m be a fxed square-free postve teger the equvalece classes of solutos of Dophate equato x + m y z form a ftely geerated abela group uder the operato duced by the complex multplcato. A bass of ths group s costructed here usg prme deals ad the deal class group of the feld Q ( m ).. Itroducto Fx a arbtrary square-free teger m >. The the set of equvalece classes of solutos of Dophate equato x + m y z () Mathematcs Subect Classfcato: R4 R F5. Keywords ad phrases: almost Pythagorea trples prme deals magary quadratc feld deal class group. Receved September 6 6 Scetfc Advaces Publshers

2 5 wll have a group structure. Ideed gve two arbtrary solutos ( a b c ) ad ( a b c ) of () we ca produce aother oe ( A B C) va the formulas A : aa mb b B : ab + ab C : cc. () Here s precse defto of the group. Cosder the set of all ordered prmtve trples ( a b c) Z Z N such that a + m b c. A trple ( a b c) s called prmtve whe the gcd ( a b c). Two trples ( a b c) ad ( A B C) satsfyg () are sad to be equvalet f there exst m Z \ {} such that m ( a b c) ( A B C) where m ( a b c) ( ma mb mc ). Ths s clearly a equvalece relato. The equvalece class of ( a b c) a b c ad the set of all such classes wll be deoted by [ ] wll be deoted by P m ad called the group of prmtve almost a b c a b c a b c Pythagorea trples. Note that [ ] [ ] but [ ] [ a b c]. Every equvalece class [ a b c] m P ca be represeted uquely by a prmtve trple ( α β γ) where α > ad hece oe could refer to prmtve trples to descrbe elemets of the group. Followg the operato () we see that for ay two classes [ a b c] [ A B C] Pm [ a b c] + [ A B C] [ aa mbb ab + ba cc]. Here s a alteratve descrpto of ths group usg a bt more techcal terms: Cosder the multplcatve subgroup say A m of ozero elemets whose orm s a square of a ratoal umber the magary quadratc feld Q ( m ). The ozero ratoal umbers wll mae a subgroup of factor group A Q s P. m Q A m ad t s easy to see that the correspodg m

3 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 53 Ths group has bee cosdered by varous authors (see e.g. [] [3] [4] ad [5]) sce the operato () was troduced by Taussy ([7]) for ordary Pythagorea trples. It s well ow that the group s ftely geerated ad has torso oly whe m 3. Descrpto of a bass has bee gve however oly for partcular values of m (see [] [3] ad [4]). The goal of ths paper s to preset a bass of the group P m usg prme deals ad the deal class group of Q ( m ) for all square-free tegers m > 3. For the proof of the followg theorem see [] or [4]. Theorem. The group each square-free >. P m s a ftely geerated abela group for m The detty elemet s [ ] [ a b c] s [ a b c] [ a b c]. Pm whe m 3. ad the verse of has elemets of fte order oly Note that Zaardo ad Zaer ([8]) studed the group cosstg of the set of equvalece classes of solutos of x y z + the rg O K of tegers of a umber feld K ad descrbed a bass for the torsofree part of that group. The group I cosder here s a proper subgroup of the oe dscussed [8]. From ow o let s assume that we have a fxed square-free teger m > 3 ad deote the magary quadratc feld Q ( m ) by K. The correspodg rg of tegers wll be deoted by O K ad the deal class group of K by Cl ( K ). The orm of a ozero deal Q of O K wll be wrtte as N ( Q). I wll deote the set of all ratoal prme umbers by P ad the set of all prme deals of O by P( O ) respectvely. K K. Ideal Classes of Order Lemma. Let c p p be the prme decomposto of a atural umber c. If assume that m ( mod 4).e. s ether ert or splts K. Suppose that for each odd prme { } p

4 54 there exsts a prme deal Q of O K s.t. N ( Q ) p ad let Q f s ert K ad Q Q f splts K (here Q s the cougate of a prme deal Q ). If the class of the there exst tegers u ad v s.t. ether u + mv ( c). Proof. If [ Q Q ] [ ] Q Q Q has order Cl ( K ) u + mv c or Q the there exsts z O K s.t. Qt t t z u + v m δ wth u v Z; δ f m ( mod 4) ad δ otherwse. If splts or ramfes K we ca tae the orm of both eds of the above equalty δ to obta 4 c u + mv. If s ert K the we apply the prevous step to the product of prme deals [ Q ] [ ] obta Q to c 4 u + mv whch mples ( c) ( u) + m( v). Sce every prme deal of O K les over a uque ratoal prme ad every prme p N les below at least oe prme deal of O K (see Theorem of [6]) we have the caocal proecto : P( O ) P whch has a lftg map l P P( O K ) such that µ l Id. : P µ K Hece f we pc such a lftg l ad compose t wth aother caocal proecto π : P( O K ) Cl( K ) we obta a map f ad the followg commutatve dagram:

5 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 55 Note. Map f does deped o the choce of the lftg l : P P( O K ). Note. Sce π s a multplcatve homomorphsm from the group geerated by P( O K ) we ca exted such map f a atural way to a multplcatve fucto from N to Cl ( K ). Note 3. A odd prme p dvdes c for a prmtve trple ( a b c) whch satsfes () f ad oly f the Legedre symbol m (see p Lemma.7 of []). It s also trval to see that f c s eve the prmtve trple above the we must have m ( mod 4) ad f m 5 ( mod 8) the 4 c. Let me ow troduce two specal subsets of Cl ( K ) ad of P respectvely. Defto. The set of all elemets of the deal class group of K whch have order at most wll be deoted by E that s E {[ A ] Cl ( K ) A s a deal of O ad [ ] [ ]}. : A K Clearly E s a subgroup of Cl ( K ).

6 56 Defto. A subset L P s defed as the subset of ratoal prmes that splt (completely) K. I the other words : { m L p P } p where m s the Legedre symbol for p > ad the Kroecer p symbol f p. Moreover we defe L P : f ( E) L where f ( E) deotes the premage of E. Clam. If m ( mod 8) ad m > 6 the L ad E Cl( K ). Proof. Sce m ( mod 8) we have the splttg of to the product of prme deals ad t s eough to show that + m m P P m + m P 4 s ot prcpal 4 O K. If oe assumes that there are x y Z s.t. P x + y m the we could fd a b Z s.t. 4 a + b m x + y m ad hece 6 ( a + mb ) ( x + my ). It s easy to see that assumptos le b or y lead to a cotradcto (for example by usg the orm) ad f by by the ( a + mb ) ( x + my ) > 6 whch s also ot allowed Note 4. If m 7 the + 5 m 5 the P ad [ 4] P. P ad [ 4] P If

7 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 57 Recall that Theorem of [4] descrbes a bass of P m for four specal cases of m whch correspod to the class umbers { } of O K. It says: Fx m { 3 5 6}. The P m s geerated by all prmtve trples ( a b p) where a > ad p s prme such that there exst u v Z wth p u + mv or p u + mv. Our ext theorem geeralzes results of [4] ad [] (cf. also wth [3]) ad gves a bass of the group P m whe every elemet of the deal class group Cl ( K ) has order at most. Approach of Baldsserr [] covered several partcular cases of m (see Theorem o page 35 of []). I all those 36 cases Cl ( K ) s ether trval or somorphc to the drect product of several copes (up to 4) of Z (the cyclc group of order ). Theorem. If L L.e. the deal class group Cl ( K ) has expoet less tha or equal to the the group P m wth m > 3 s geerated freely by the set of all prmtve trples ( a b p) or ( a b p) where p L ad a ad b are postve tegers such that a + mb p or p a + mb ( ). Proof. Frst let s prove that f p L the there exsts a uque coprme par ( a b) N N s.t. ether a mb p p + or a + mb ( ) but ot both. If p L the from our Lemma we deduce that there exst tegers u ad v s.t. ether u + mv p or + mv u ( p ). If we assume that p s odd ad there are two pars of relatvely prme tegers ( a b) ad ( u v) s.t. p a + mb p ad at the same tme u + mv ( ) we cosder two ew elemets of P : m [ u v p] ± [ a b p] [ au mbv av + bu p ] [ au mbv av + bu p ].

8 58 Sce ( bu + av)( bu av) b u + b mv ( a v + mb v ) p ( 4b v ) f v ± b we have p bu + av or p bu av or both. I the last case we d have p bu whch s mpossble for a odd prme ad two pars of relatvely prme tegers ( a b) ad ( u v). If p dvdes oly bu + av the p bu + av ad hece au mbv p.e. [ u v p] + [ a b p] [ s t ] for some tegers s ad t. Sce m > 3 we deduce that t ad av bu whch cotradcts the assumptos. If p dvdes oly bu av a smlar argumet leads to the cotradcto as well. If v ±b we aga have a cotradcto wth ( u v). Aalogously oe ca cosder the remag cases whe p s odd ad prove the uqueess of a coprme par ( a b) N N s.t. ether a mb p we could have + mb 6 [ 4] P. 5 + p or a + mb ( p ). If a oly for [ 3 4] P7 or for τ Secodly we prove that the set of all such prmtve trples ( a b p) wth p L ad τ { } geerate the group P m. Tae ay prmtve [ a b c] P m ad assume that the prme decomposto of c s c p p where ad p are dstct odd ratoal prmes. The by Note 3 ad our Lemma for each { } there exst u mv u v N N s.t. + or u + mv ( p ) that s ether [ u v p ] or [ u v p ] s a geerator. Assume wthout loss of geeralty that > ad cosder p τ [ a b c] ± [ u v p ] τ [ au mbv av + bu c p ] τ [ au mbv av + bu c p ] (3) wth τ { }. As above we have ( bu + av )( bu av ) ( mod p ) ad hece p dvdes at least oe of ( bu + av ) or ( bu av ) or p dvdes each of these umbers. Assumg the last possblty we'll get a

9 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 59 cotradcto wth the fact that our trples o the left had sde of (3) are prmtve. Assumg the frst possblty we obta that p au mbv. The we have for D ( au mbv ) p ad E ( av + bu ) p c p τ τ [ a b c] [ u v p ] + [ D E ] whch mples that there exst tegers γ ad ω such that [ a b c] ± r [ u v p ] + [ γ ω ] where τ { } ad r { } for each { } ad. Accordg to our Note 3 [ γ ω ] [ ] τ uless m ( mod 8). I the latter case Clam guarates that m { 7 5} ad oe ca easly prove usg ducto ad applyg the dvsblty argumet we ust used above to the elemets [ γ ω ] ± [ q r 4] + [ qγ mrω qω + rγ ] + [ qγ mrω qω + rγ ] that [ γ ω ] ± ( ) [ q r 4] where [ q r 4] [ 3 4] f m 7 ad [ q r 4] [ 4] f m 5 respectvely. Thrdly we show that the geeratg set s free of ay otrval relatos. Suppose that there exsts a prmtve trple [ K L M ] Pm wth two dfferet presetatos by elemets from the geeratg set.e. z t J t T [ K L M ] z (4) τ where each z [ a b p ] wth τ { } ad J T s a prmtve trple wth p L. We ca assume wthout loss of geeralty that J T /. Sce the thrd compoets get multpled whe we add

10 6 two elemets of P m we ca wrte the thrd compoets of the left ad rght had sdes of (4) (before reducg to the correspodg prmtve trples) as t p p t J t T ad respectvely. Sce the left ad rght had sdes of (4) are equal ad sce J T / we see that M. The we deduce from our Note 3 above that [ L M ] [ ] K uless m ( mod 8). Suppose ow that m ( mod 8) ad the thrd compoet of oe of z s dvsble by a odd τ prme p. The we could wrte the left had sde of (4) as [ a b p] + [ U V W ] where ad { }. ε [ α β p ] τ τ Wrtg [ a b p] as (see Lemma below) we ca rewrte the left had sde of ε (4) as [ α β p ] + [ U V W ] ε where the trples ( α β p ) ad ( U V W ) are ot ecessarly prmtve but ( p W ) ad ( β p ). Thus ε ε [ α β p ] + [ U V W ] [ αu mβv αv + βu p W ] [ K L ] whch mples that p α U mβv ad also p α V + βu. I such stuato we would also have p W whch s ot allowed. Thus the left ad rght had sdes of (4) could oly have group elemets where the thrd compoet s a power of. But there s oly oe such bass elemet whch s [ 4] ad hece we ca ot have two dfferet otrval presetatos of [ K L M ]. Lemma. Suppose for a odd prme p we have a prmtve trple [ u v p b w] Pm s.t. ( w) b [ u v w ] [ u v p w] wth ( ) p ad b. Tae ay N. If u the p. v b w

11 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 6 b Proof. Frst observe that [ u v w ] [ u mv uv p w ] ad b f p does ot dvde w we must have p uv whch s ot allowed. Let b us wrte u mv ( p w ) mv p A mv for some A. The oe ca use ducto to show that [ u v w ] [ p A ± m v Z p B ± m u v b p w ] wth both A ad B. Aalogously oe proves a smlar formula Z b whe s odd where the secod compoet of ( + ) [ u v p w] wll be v p B ± m v. Clearly both cases these formulas mply that p b w. Example. Let m 35 the Cl( K ) C ad we have L L { }. Here are frst few geeratg trples whch satsfy equato a + mb p : [ 7] [ 7 73] [ 43 83] [ 3 49] { } ad here are several frst geeratg trples satsfyg equato a + mb ( p) : {[ 3] [ 3 3 ] [ 9 3 3] [ 9 3 7] [ 3 9 9] [ ] [ ] }. 3. Ideal Classes of Hgher Order From ow o I wll assume that Cl ( K ) has elemets of order hgher tha ad hece E wll be a proper subgroup of Cl ( K ). I such a case we ca exted the dagram defg map f to the followg oe:

12 6 Here map ρ s the caocal proecto oto the factor group Cl ( K ) E ad the map g : ρ f. Sce Cl ( K ) s fte we ca preset the factor group Cl ( K ) E as a drect sum of cyclc groups Cl ( K ) E G G G wth correspodg orders deoted by h : G { }. For each G pc a deal P P( O ) s.t. G ρ π( ) (c.f. wth the K P presetato of the factor group Cl ( S) M as a drect sum of cyclc groups ad the correspodg costructo rght after that o page 84 of [8]). Now let s use the lftg l : P P( O K ) to costruct a map β : L Pm as follows. Frst tae ay prme p L L. \ Case. Suppose that µ ( P ) p for some { }. The h ( π ( P )) δ E ad by Lemma above we ca wrte ( p ) u + mv for some postve tegers u ad v (where as above δ { } ). If we suppose that p gcd ( u v) we wll deduce that P (the cougate deal of P ) dvdes O a power of P whch s mpossble therefore we ca assume that K gcd ( u v). Thus we obtaed a prmtve almost Pythagorea trple h (5) δ h ( u v p ) defe whch s uque up to sgs of u ad v. Therefore we h δ β( p) : [ u v p ] where ( u v) N N.

13 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 63 Case. Suppose further that p L\ ( L { µ ( P ) { }}). The for the prme deal P l( p) there exst oegatve tegers a { h } such that p a p P P a P p a E ad hece P P p ap P u + v m δ (6) for some ( u v) Z Z. Clam. u gcd δ v δ. Proof. Suppose that we have a prme q whch splts ad dvdes both u ad v. The f Q : l( q) we have a p a q Q Q P P P p. But ths s mpossble sce oe of the deals { P P P } has order Cl ( K ) ad they all have dfferet orms whle N ( Q) N ( Q ) q. Thus as the frst case we foud a relatvely prme par of tegers ( u v) s.t. δ a αp u p + mv ( p p p ). Note that ths tme the par ( u v) s ot ecessarly uque. If for t δ t example h ad we foud a trple ( u v p p ) the for h δ the trple ( u v p ) foud Case above oe of the trples δ t δ h [ u v p p ] ± [ u v p ] would be aother prmtve almost Pythagorea trple wth the thrd compoet δ t p p ad the frst two compoets dfferet from ± u ad ± v. We ca alterate the costructo gvg (6) ad show that each

14 64 h a p may be tae from the set { }. a > h p we ca multply the product h a p h a p ( P P ) p ad obta a elemet Ideed f for example a p a P p P P by b ( ) wth h P P p ap P E b p h a p. Now we are ready to defe map P β : L \ L m for p L \ ( L { µ ( ) { }}). Tae ay such prme umber p the as we P ust explaed above there exst relatvely prme tegers u ad v such a δ p p that u + mv ( p p p ) ad { a p a p } are the coordates of the deal P l( p) Cl ( K ) E wrtte the bass { ρ π( P ) or ρ π( P ) ρ π( P ) or ρ π( P )} whch meas that h δ a p a p for each. Amog such prmtve trples ( u v p p a p p ) tae the oe wth the smallest value of u say u ad defe δ a : p a β( p) [ u v p p p p ] where ( u v ) N N. We ca exted map β to elemets of L as well by defg β ( p ) : [ a b p] + ad ( p) : [ a b p] f a mb p whe p L. Thus we obta a map β : L Pm. a β f a + mb ( ) p Note 5. It s clear from the costructo that ths map β : L P s m oe-to-oe. It s also clear that β depeds o several choces we ve made the costructo ad partcular t depeds o the choce of the prme deals proectg oto the geerators of subgroups G of Cl ( K ) E.

15 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 65 Example. Let m 3 the Cl( K ) C3 ad we have L { }. Here L { } L wth the correspodg values β ( 59 ) [ 3 59] β( ) [ 83 ] β( 67) [ 4 67] β ( 73 ) [ 36 73]. If we choose the deal + 3 lyg over as the oe whch gves the geerator P of Cl ( K ) E C 3 the β ( ) [ 7 3 ] ad also β ( 3) [ ( 3 ) ] β( 3) [ 9 9 ( 3 ) ] β( 9) [ 9 5 ( 9 ) ] ad so o... If we choose the deal lyg over 3 as the oe whch gves the geerator of Cl ( K ) E the β ( 3) [ ] ad also C 3 β ( ) [ ( 3) ] β( 3) [ 7 8 ( 3 3) ] β( 9) [ 4 6 ( 9 3) ] ad so o.... Usg ths map β we ca ow descrbe a bass of the group 3 3 P m for all m > 3 whe the correspodg deal class group Cl ( K ) has elemets or order hgher tha (cf. wth Theorem 3 of [8]). m Theorem. Im ( β) Pm forms a bass of the free abela group P m > 3. Proof. Frst I expla why Im ( β) s a geeratg set. Tae ay prmtve [ a b c] Pm the deal ad wrte the prme decomposto O K for a b m l l l T T T r r : T ad ote that l sce ( a b).

16 66 If T s ot oe of the chose geerators { P P P } for Cl ( K ) E the usg the costructo we used Case we ca wrte that b t bt bt T Q Q Q where meas the equalty up to a product by a elemet of E t N( T ) ad each Q { P P } so that h b t b b t { }. Deote the product t bt T Q Q Q by T ad we obta T T l T l T lr r b b b t ( t t Q Q Q ). l Smlarly we ca elmate from T all T whch are ot the set { P P P }. Suppose that o the other had we have T P (we ca always reame T f eeded). The usg dvso wth remader we ca wrte l q h + r wth r < h ad hece ( T ) T Q where Q ad T ( P ) whch also would gve us P h l q r T T q l l r T T r ( Q ). r Therefore f we let ω q whe P ω l otherwse we deduce that T ad l q h + r ad ω ωr T r E T T sce each T E ad T s prcpal. Hece we ca wrte for the cougate deal a + b m ω ω T T r r I for some I E. Sce each l s eve ths last equalty mples [ a b c] γ β( t ) + γ β( t ) + + γ β( t ) + [ g h d] wth all γ r r Z ad where d ca be dvded oly by prmes from the set L ad h powers of ( p ) for { } (recall that by our choce Cl ( K ) E

17 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 67 s geerated by ρ π( P ) where µ ( P ) p ad h s the order of the correspodg cyclc subgroup G ). Repeatg the secod step from the proof of Theorem above ad presetg [ g h d] as a lear combato of elemets from β ( L ) ad from { β ( p ) β( p )} we coclude that P m s geerated by the mage Im( β ). To show that the set Im ( β) s free of ay otrval relatos I wll use the same approach that was used the thrd part of the proof of Theorem. Let s suppose that there s a otrval relato amog the geerators.e. there exsts a prmtve trple ( K L M ) s.t. [ K L M ] Pm ad I s β( p ) [ K L M ] t β( p ) ad I J / (7) J where { p I } { p J } L ad s Z for all I ad J. If we assume that there exsts a odd prme q { p I} whch does ot le below ay of the chose delas P P mappg to the geerators of Cl ( K ) E the usg our Lemma we ca wrte the left had sde of (7) as s β I p q t ( p ) + s β( q) [ K L M ] + [ u v q w] where the trples ( K L M ) ad ( u v q w ) are prmtve. It follows from the defto of operato P m fact that I J / ad our costructo of map β that ( M q) ad also ( M q). But s s [ K L M ] [ K L M ] + [ u v q w] [ uk mvl vk + ul q w ] M s whch s possble oly whe q dvdes both uk mvl ad vk + ul. s I such case q wll also dvde vk + mvl v M whch s possble oly f s. Ths proves that f a otrval relato exsts amog elemets of Im( β ) t could oly volve elemets from β ( ) ad from { β ( ) β( p ) β( p )}. s s L

18 68 If we assume that there s a otrval relato (7) wth { p I } { p J } L { p p } ad for example that odd p { p I } the usg the defto of map β ad Lemma we ca rewrte the left had sde of (7) ths tme as s I p p β s h ( p ) + s β( p ) [ K L M ] + [ u v p w] where ( K L M ) ad ( u v p w ) are prmtve trples such that s h ( M p ). Sce p { p } we also have ( M p ). We wll J come to a cotradcto aga usg exactly the same argumet we ust used above by presetg [ K L M ] p s h elemets from { β ( p ) β( p )} relato ether (wth oly possble excepto of β ( ) ). as [ w M ]. Hece the ca ot be volved a otrval Fally assumg that there s a otrval relato amog elemets of { β ( p) p L {} } oly oe ca repeat the thrd part of the proof of Theorem to deduce that the left ad rght had sdes of (7) could oly have group elemets where the thrd compoet s a power of. But there could be oly oe such bass elemet whch s β ( ) ad hece we ca ot have two dfferet otrval presetatos of [ K L M ]. Ths fshes the proof of Theorem. Here s a example whch motvates ad llustrates the approach I used above the costructo of map β. Example 3. Let m 974 the Cl ( K ) C C 3. Choose the deal 5 + m as the geerator P of C ad the deal m as the geerator P of C 3. The ρ π( P ) ad ρ π( P ) wll be the chose geerators of the factor group Cl ( K ) E C 6 C 3. Usg a computer oe ca easly fd that subset L starts wth umbers

19 A BASIS OF THE GROUP OF PRIMITIVE ALMOST 69 { } L ad correspodgly ( 937 ) [ ] β ad β ( 983 ) [ ]. O the other had the set L cotas may smaller prmes as well: L { }. 6 For the deal P we have P ad 3 3 β ( 4 ) [ 69 4 ] wth ( 4 ). + For the deal P we have P ad β ( 5) [ ]. Now let s choose a prme whch s ot L { µ ( P ) µ ( )} for P example p 3. The we have the product P P P E sce m m ad Hece β( 3 ) [ ]. Notce also that we have two prmtve trples for p 37 where the thrd compoets are the same: 3 3 [ ] wth ( P ) ad 3 3 [ ] wth ( P ) Sce we have [ ] + [ ] [ ] ad hece accordg to our defto we have β ( 37) [ ]. 3

20 7 Refereces [] N. Baldsserr The group of prmtve quas-pythagorea trples Red. Crc. Mat. Palermo () 48() (999) [] D. Cox Prmes of the form x + y Fermat Class Feld Theory ad Complex Multplcato A Wley-Iterscece Publcato Joh Wley & Sos Ic. New Yor 989. [3] E. Ecert The group of prmtve Pythagorea tragles Math. Mag. 57() (984) -7. [4] N. Krylov ad L. Kulzer The group of prmtve almost Pythagorea trples Ivolve 6() (3) 3-4. [5] F. Lemmermeyer Hgher Descet o Pell Cocs III The Frst -Descet preprt 3. [6] D. Marcus Number Felds Uverstext Sprger-Verlag New Yor-Hedelberg 977. [7] O. Taussy Sums of squares Amer. Math. Mothly 77 (97) [8] P. Zaardo ad U. Zaer The group of Pythagorea trples umber felds A. Mat. Pura Appl. (4)59 (99) g