ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES

Transcription

1 Journal of Algebra, Nuber Theory: Advances and Applcatons Volue 3, Nuber, 05, Pages 3-8 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES Feldstrasse 45 CH-8004, Zürch Swtzerland e-al: Abstract We express the nuber of dstnct prtve Pythagorean quntuples n ters of the total nuber of prtve representatons of a square as a su of four squares (countng zeros, perutatons, and sgn changes), two twsted Euler functons wth Drchlet characters of perod four and eght, and three countng forulas for bnary sus of squares.. Introducton The representaton of ntegers by sus of squares has attracted the attenton of any atheatcans fro Dophantus, Euler, Lagrange, to Gauss and any others. One of the oldest classes of such representatons concern Pythagorean trples. More generall for fxed n 3, Pythagorean n-tuples concern the representatons of squares by sus of n nonzero squares. In densons n 5 countng forulas for the do not see to exst. Moreover, t s only recently that Hürlann [5] has derved an exact and an asyptotc forula for the nuber of prtve Pythagorean quadruples. 00 Matheatcs Subject Classfcaton: E5, A5, B34, D45. Keywords and phrases: Dophantne equaton, su of squares, ternary quadratc for, arthetc functon, twsted Euler functon. Receved March 7, Scentfc Advances Publshers

2 4 A prtve Pythagorean quntuple s descrbed by a soluton n non-zero natural nubers w, x, z, t of the Dophantne equaton w + x y + z + satsfyng the condton gcd ( w, x, z, t) =. Generalzng the countng echans n Hürlann [5], we derve an exact countng forula for the nuber of prtve Pythagorean quntuples. A ore detaled account of the content follows. Secton suarzes soe basc facts about the representaton of nubers by sus of squares and ore generally by quadratc fors. Usuall a countng forula for the nuber of representatons by a quadratc for ncludes zeros, perutatons, and sgn changes. If one counts prtve representatons onl then Möbus nverson of such a countng forula ust be perfored. To count Pythagorean n-tuples, t s further necessary to dsregard fro countng zeros. The algebrac echans for ths s explaned and a basc reducton forula to count the nuber of prtve quntuples s derved n Equaton (.8). The Sectons 3 and 4 deterne countng forulas for nvolved ternary and bnary squares representatons. The an forula for the nuber of dstnct prtve Pythagorean quntuples s obtaned n Theore 5. and llustrated n the Tables 5. and 5... The Basc Algebrac Countng Mechans In classcal arthetc, one s nterested n representatons of a nuber as a su of k squares such that x + x + + xk =. The nuber of such representatons, countng zeros, perutatons, and sgn changes, s denoted by r k ( ). Slarl the abbrevaton R k ( ) denotes the nuber of prtve representatons of as a su of k squares countng zeros, perutatons, and sgn changes. Gven a forula for r k ( ), one obtans fro t a forula for R k ( ) through Möbus nverson of the basc dentty (e.g., Grosswald [6], Theore, Subsecton.)

3 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN 5 ( ) = rk Rk. (.) d d Ths technque has been largely exploted by Cooper and Hrschhorn [], who obtan a wde varety of forulas for R k ( ) ncludng the range k 8 for any, and the range 9 k for certan values of. The nuber of dstnct prtve representatons of as a su of k k non-zero squares such that x + x + + x = wth x 0, whch s denoted by Rk d ( ), has not been studed very often. It s only recently k j= j that Hürlann [5], Lea 4, has derved a forula for R3 d ( t ), nuber of prtve Pythagorean quadruples. the Möbus nverson s not restrcted to sus of squares. Gven a quadratc for Q ( x, x,, x ), one s n general nterested n nteger k solutons of the Dophantne equaton Q ( x, x,, xk ) = for gven. Then, f one denotes by r Q ( ) the total nuber of solutons, and by R Q ( ) the total nuber of prtve solutons, one has the generalzed forula ( ) = rq RQ. (.) d d Möbus nverson of (.) yelds the forula (Cooper and Hrschhorn [], Equaton (.3)) ( ) = ( ) + RQ rq rq rq p p p p p, p, p3 p, p n ( ) r Q + +, rq (.3) p p p p p 3 p,, pn n where p,, p, pn are the dstnct pres whose squares dvde.

4 6 Let us return to the topc of the present contrbuton. Bachet (58-638) conjectured n 6 that every natural nuber can be wrtten as a su of four squares. The frst coplete proof of ths stateent s due to Lagrange [7]. Soe further hstorcal developent and concrete expressons for r 4 ( ), respectvel R 4 ( ), are found n Cooper and Hrschhorn [] (the forulas are found n Lea 3, respectvel Theore ). An exact forula for the nuber R d ( ) of prtve Pythagorean quntuples has not been derved so far. As wll be proved later, there exst prtve Pythagorean quntuples f, and only f, the nuber t > 3 s odd or t s exactly dvsble by. Therefore, Pythagorean quntuples for even nubers t 4 dvsble by 4 are always ultples of prtve ones. Let us explan the basc echans used to derve a forula for R d ( ). Consder the R ( ) nteger quadruples ( w, x, z) solvng 4 t w x + y + z 4 t + takng nto account zeros, perutatons, and sgn changes. The solutons are sad to be prtve f these quadruples satsfy the condton gcd ( w, x, z, t) = n the sense that any nteger s a dvsor of zero, a conventon ade throughout. Lookng at possble zeros and equal entres the dstnct prtve solutons can take 7 dfferent fors, nael ( w, x, z), ( w, x, y), ( w, x, x, x), ( w, w, y), ( 0, x, z), ( 0, x, x, z), 4 t and ( 0, 0, y, z), wth dstnct entres w, x, z 0. For each for, one ust deterne the nuber of resultng representatons countng perutatons and sgn changes, as well as the nuber of dstnct prtve solutons generated by ths for. The latter countng functon s denoted by D ( t ), where stands for the quadratc for type of the correspondng Dophantne equaton. The requred nforaton s suarzed n Table..

5 Table.. Fors, Dophantne equatons, representatons and dstnct solutons for R ( ) 4 t # of # of sgn total # of # dstnct For Dophantne equaton perutatons changes representatons solutons ( w x, z), w x + y + z ( w x, y) 4 t D ( ), w x + y ( w x, x, x), w 3x ( w w, y), w y (, x, z) 0 x y + z (, x, y) 0 x y (, 0, z) 0 y z,, t D ( )( ),3 t D ( )( ), t D ( )( ) 3 t D ( ), t D ( )( ) t D ( )

6 8 4 t By defnton of the countng functon R ( ), one has the dentty 384D 4 ( t ) + 9D(,, )( t ) + 64D(, 3)( t ) + 96D(, )( t ) + 9D ( t ) + 96D( )( t ) + 48D ( t ) = R ( ). (.4) 3, 4 t Addtonall by defnton of the frst 4 fors, one has the dentty d ( t ) + D( )( t ) + D( )( t ) + D( )( t ) = R ( ). (.5) D4,,, 3, 4 t On the other hand, denote by ( ) ( t ),, R d the nuber of non-zero dstnct prtve solutons of the Dophantne equaton w + x + y, and by R d ( ) 3 t the nuber of non-zero dstnct prtve solutons of x + y + z. Snce the dstnct prtve solutons defnng R3 d ( t ) are of the three dfferent fors ( x, z), ( x, y), and ( 0, y, z), wth dstnct entres x, z 0, one obtans slarly to (.5) the dentty d D ( t ) + D( )( t ) = R ( ). (.6) 3, 3 t To obtan an dentty for R ( d ) ( t,, ), one consders slarly to Table. the dfferent fors assocated to the countng functon denoted R (,, )( t ) (the nuber of prtve solutons of x y + z + ) as suarzed n Table.. Though only partally requred here, the whole nforaton wll be needed n Secton 3.

7 Table.. Fors, Dophantne equatons, representatons and dstnct solutons for R ( )( ),, t For Dophantne equaton ( x z), x + y + z ( x y), x + 3y ( x x, z), x + z ( z z, x), z + x (, 0) x x + y (, z) 0 y + z # of perutatons # of sgn changes total # of representatons # dstnct solutons D (,, )( t ) D (,3)( t ) D (, )( t ) D (, )( t ) D ( t ) D (, )( t )

8 0 It s portant to note that the fors ( x, x, z) and ( z, z, x), though solutons of the sae Dophantne equaton, generate dstnct solutons of x + y + z. Therefore, by defnton of the frst 4 fors n Table., one obtans the dentty d,, D(,, )( t ) + D(, 3)( t ) + D(, )( t ) = R ( ) ( t ). (.7) 4 3 t,, t Now, solve (.5)-(.7) for D ( t ), D ( ), and D ( )( ), respectvel and nsert nto (.4) to get after straghtforward algebra the basc relatonshp d d,, ( ) ( ) ( ) ( ) ( ) 384R4 t = R4 t + 9R t 9R3 t + 8D(, 3)( t ) 96D(, )( t ) + 96D(, )( t ) 48D ( t ). (.8) Once forulas for the rght-hand sde quanttes have been deterned, the left-hand sde, whch yelds the nuber of prtve Pythagorean quntuples, wll also be deterned. 3. Countng Forulas for the Ternary Squares Representatons A forula for R3 d ( t ), whch deternes the nuber of prtve Pythagorean quadruples, s found n Hürlann [5], Lea 4. It s gven by ( ) R3 d t = ϕ( t, χ4 ) D ( t ) + D(, )( t ), (3.) 8 where t 3 s odd, and ϕ( t, χ4 ) ( χ4 ( p) / p) s the twsted p t Euler (totent) functon wth Drchlet character of perod four defned by 0, p =, χ4 ( p ) =, p ( od 4), (3.), p 3 ( od 4). d

9 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN Applyng the sae algebrac countng echans as n Secton, we derve now a forula for R d ( ) ( t ). Fro Table., one obtans,, slarly to (.4) the relatonshp 6D (,, )( t ) + 6D(, 3)( t ) + 6D(, )( t ) + 8D ( t ) + 8D(, )( t ) = R(,, )( t ). (3.3) Solvng (.7) for D ( ) ( ) and nsertng nto (3.3), one obtans the,, t forula ( ) ( ) 6R d t = R(,, )( t ) + 6D(, )( t ) 8D( t ) 8D(, )( t ). (3.4),, To obtan a forula for R (,, )( t ), one apples Möbus nverson to the countng forula for r ( ) ( ) by Cooper and La [3]. For the postve,, t e λ nteger t 3, let t = p be ts unque pre factorzaton. Then, one has = λ + p p r(,, )( t ) = 4b( e ), p = p p λ (3.5) where, f e = 0, b ( e ) = (3.6) 3, f e, and the values of the Legendre sybol are gven by p =,, f f p or 3 p 5 or 7 ( od 8), ( od 8). (3.7) Fro ths, one gets through applcaton of the Möbus nverson forula (.3), the followng forula for the nuber of prtve solutons of the equaton x + y + z :

10 λ 4 = ϕ( χ ) ( ) ( )( ) = p p 4 t, 8, f t,, 3 od 4, R,, t p = 0, f t 0 ( od 4), (3.8) where ϕ( t χ ) ( ( p) / p) s the twsted Euler (totent), 8 χ8 p t functon wth Drchlet character of perod eght defned by ( ), f p or 3 od 8, χ8( p ) = (3.9), f p 5 or 7 ( od 8). We are ready for the followng result. Proposton 3. (Nuber of non-trval dstnct representatons of squares by the ternary quadratc for forula x + y + z ). One has the ( ) ( t ) =,, R d ϕ( t, χ ) ( ) ( )( ) 8 D t D, t, f t > 4 ϕ( s, χ8 ) + D( s ), f t = s, s > odd,, f t =, 0, f 4 t. s odd, (3.0) Proof. Ths follows fro (3.4) usng (3.8). If t > s odd, then one, = has D ( )( t ) 0 because the equaton If t = s wth s > odd, one sees that x y + has no soluton. D (, )( t ) = D( s ) = D ( s ). (3.) Moreover, one has D ( s ) 0 (see Cooper and Hrschhorn [], Theore, 4 = Equaton (.6)) and D ( ) ( s ) 0. The latter s seen as follows. If, 4 =

11 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN 3 x + y = 4s, then necessarly x = X, hence X + y = s and y. But then gcd( x, t), whch contradcts the condton gcd ( x, t) =. Slarl f 4 t, then all ters n the rght-hand sde of (3.4) vansh. The result s shown. 4. Countng Forulas for the Bnary Squares Representatons Accordng to the basc relatonshp (.8) forulas for the nuber,, t of bnary squares representatons D ( t ), D( )( t ), D( )( ), and D (, 3)( t ) ust be deterned. Agan, we suppose that the postve e nteger t has the pre factorzaton λ t = p. The frst two = countng functons are well-known and gven by D ( t ) =, f p ( od 4), =,,, e = 0, otherwse, 0, (4.) respectvel D(, )( t ) =, f p, 3 ( od 8), =,,, e = 0, otherwse. 0, (4.) A reducton forula for D ( )( ) s contaned n the proof of Proposton 3.. One has, t D( s ), f t = s, s odd, D (, )( t ) = (4.3) 0, otherwse. It reans to deterne the nuber of dstnct prtve solutons of the equaton x + 3y. Agan, one frst fnds a forula for the nuber of all representatons r ( ) ( ) takng nto account zeros, perutatons,, 3 t and sgn changes, and then apply Möbus nverson to get an expresson

12 4 for R (, 3)( t ), fro whch one gets D (, 3)( t ) = R(, 3)( t ). Accordng to 4 Dckson [4], Exercses XXII, no. 3, p. 80, Secton 5 (see also Berndt [], Theore 3.7.5, p. 75), one has for an odd nuber s : E( s), f k = 0, k r (, 3)( s) = 6E() s, f k s even, (4.4) 0, otherwse, where ( n) = d ( n) d ( n) s the excess of the nuber of dvsors E, 3, 3 ( od 3) of n over the nuber of dvsors ( od 3) of n. Based on the e pre factorzaton t = p, one obtans after soe calculaton the forula = λ, f p ( od 3), =,,, e D(, 3)( t ) =, f p ( od 3), =,,, e 0, otherwse. = 0, =, (4.5) 5. The Exact Nuber of Prtve Pythagorean Quntuples Up to the countng functon R 4 ( t ), all ters on the rght-hand sde of the basc relatonshp (.8) have been deterned n the Sectons 3 and 4. For the reanng ter, one borrows fro Cooper and Hrschhorn [], Theore, Equaton (.7), the expresson 8t ( + / p), f t s odd, p t R4 ( t ) = 6s ( + / p), f t = s, s odd, p s 0, otherwse. The an result of the present contrbuton follows. (5.)

13 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN 5 Theore 5.. The nuber of dstnct prtve Pythagorean quntuples s deterned as follows: t ( / p) ( t, 4 ) ( t, ) + ϕ χ + ϕ χ d R4 ( t ) = p t, f t > s odd, D( t ) D(, )( t ) D(, 3)( t ) d R4 ( ) ( / ) (, ) ( ) (, 3)( 4 t = s 8 ), 4 + p + ϕ s χ + D s + D s p s (5.) f t = s, s > odd, (5.3) R4 d, f t = ( t ) =. 0, f 4 t (5.4) Proof. We dstngush between the two an cases. Case. t > s odd., = Snce D ( )( t ) 0 by (4.3), one gets through nserton of (3.) and (3.0) nto (.8) that 384R4 d ( t ) = R4 ( t ) + 9{ ϕ( t, χ8 ) D ( t ) D(, )( t )} 4 9{ ϕ( t, χ4 ) D ( t ) + D(, )( t )} 8 + 8D(, 3)( t ) + 96D(, )( t ) 48D ( t ) = R4 ( t ) 4ϕ( t, χ4 ) + 48ϕ( t, χ8 ) 48D( t ) 96D (, )( t ) + 8D(, 3)( t ). Insertng (5.) and dvdng by 384 one obtans (5.).

14 6 Case. t = s, s > odd. R3 d = In ths stuaton, one has ( t ) 0 (no prtve cubod wth even dagonal), as well as D ( t ) = D( )( t ) 0 and D ( )( t ) = D ( ) by, =, s (4.)-(4.3). Insertng ths and (3.0) nto (.8), one has 384R4 d ( t ) = R4 ( t ) + 9{ ϕ( s, χ8 ) + D( s )} + 8D(, 3)( t ) 96D( s ) = R ( t ) + 96ϕ( s, χ ) + 96D ( s ) + 8D( )( ). 4 8, 3 t Insertng (5.) and dvdng by 384 one obtans (5.). The exact calculaton of all ters n (5.)-(5.3) requres a factorzaton table of the dstnct pre factors of all odd nubers. To llustrate, we have calculated the nuber of prtve Pythagorean quntuples for all t < 000. Table 5. provdes the detaled cuulatve count for all t < 00 and Table 5. a suarzed count.

15 Table 5.. Cuulatve nuber of prtve Pythagorean quntuples below 00 Odd t Cu. Odd t Cu. Odd t Cu. Odd t Cu. Even t Cu. Even t Cu

16 8 Table 5.. Cuulatve nuber of prtve Pythagorean quntuples below 000 Lt Odd t Even t = s Total References [] B. C. Berndt, Nuber Theory n the Sprt of Raanujan, Student Matheatcal Lbrar Vol. 34, Aercan Matheatcal Socet Provdence, RI, 006. [] S. Cooper and M. D. Hrschhorn, On the nuber of prtve representatons of ntegers as su of squares, Raanujan Journal 3 (007), 7-5. [3] S. Cooper and H. Y. La, On the Dophantne equaton n = x + by + cz, Journal of Nuber Theory 33 (03), [4] L. E. Dckson, Introducton to the Theory of Nubers, Unversty of Chcago Press, Reprnt (957), Dover Publcatons, 99. [5] W. Hürlann, Exact and asyptotc evaluaton of the nuber of dstnct prtve cubods, Journal of Integer Sequences 8(), Artcle (05). [6] E. Grosswald, Representatons of Integers as Sus of Squares, Sprnger, New York, 985. [7] J. L. Lagrange, Déonstraton d un théorèe d arthétque, Nouveaux Méores de l Acad. Royale des Scences et Belles Lettres de Berln, (Oeuvres 3 (770), ). g