Quartic residuacity and the quadratic character of certain quadratic irrationalities

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1 Intrnational Journal o Numbr Thory Vol. 11, No c World Scintiic Publishing Comany DOI: /S Quartic rsiduacity th quadratic charactr o crtain quadratic irrationalitis Ronald J. Evans Dartmnt o Mathmatics Univrsity o Caliornia San Digo La Jolla, CA 92093, USA rvans@ucsd.du Knnth S. Williams School o Mathmatics Statistics Carlton Univrsity Ottawa, ON, Canada K1S 5B kwilliam@connct.carlton.ca Rcivd Fbruary 201 Acctd 5 Fbruary 2015 Publishd August 2015 W rov a gnral thorm that valuats th Lgndr symbol A+B m undr crtain conditions on th intgrs A, B, m th rim. Th valuation is in trms o aramtrs aaring in a binary quadratic orm rrsnting. Ththormhas alications to quartic rsiduacity. Kywords: Quadratic charactr; quartic rsiduacity; irrationalitis; undamntal unit; discriminant; binary quadratic orms; Gauss s thory o gnra. Mathmatics Subjct Classiication 2010: 11A15, 11E25 1. Introduction Various Lgndr symbols o th orm A+B m hav bn valuatd in th litratur; s or xaml [, 9, 11, 13, 17 20, 22]. For a survy on th dtrmination o such symbols, s [10]. Th main rsult o our ar is a gnral thorm that valuats a larg class o Lgndr symbols A+B m in trms o aramtrs occurring in binary quadratic orms rrsnting th rim. Our thorm is vry dirnt rom thos in th ars citd abov. In this sction, w stat our thorm giv an alication to quartic rsiduacity. In Sc. 2, w rov th thorm. Thn in Sc. 3, w rsnt intrsting xamls as corollaris. Scial cass o som o our corollaris mbrac rsults in th litratur that wr obtaind by dirnt mthods. Corollaris 11 1 rovid 287

2 288 R. J. Evans & K. S. Williams ininit classs o xamls o our thorm, as wll as ininit classs o quartic rsiduacity critria. Our thorm maks us o th Kronckr symbol x Kx, y :, y as dind in [3,. 28] or arbitrary intgrs x, y. Rcall that th Kronckr symbol coincids with th Jacobi symbol whnvr th lattr is dind. For x, y 1,th gnralizd quadratic rcirocity law x yk 1,y 1.1 y x holds i only i x y ar not both ngativ [3,.].Wwillndth ollowing riodicity rorty or th numrator : x + yz x w z x, y, whn z>0x, z 1, 1.2 z z whr { 1 i 2 z, y, x + yz/2, w z x, y othrwis. This is asily rovd by actoring th dnominator z intoaowrotwotims an odd intgr. A usul ormula or w z x, y isgivnby w z x, y 1 y2x+yz/, whn 2 z. 1. Thorm. Lt δ {±1, ±2} lt a, b, c, d, r, s t b nonzro intgrs such that acr 2 bds 2 δt a>0, d > 0, r > 0, s > 0, a, s d, r r, s Lt b an odd rim such that acδ bdδ 1, t, 1.7 so that abcd can b takn as an intgr whos squar quals abcd modulo. Suos that whr ar intgrs such that cd 2 ab 2, 1.8 >0, >

3 Quartic rsiduacity th quadratic charactr 289 Din A B by A : rc + sb, B : sd + ra Thn rac + s abcd ck 1, δk 1,ad rk 1,a B d c s a w cd 2, abw B ada 2, bcbw a sd, r w d ra, s, 1.11 indndnt o th choic o squar root o abcd modulo. Whil th right sid o 1.11 dos not look articularly siml, many choics o th aramtrs yild lgant valuations. This is illustratd in Sc. 3. By , w hav rac + s abcd rac s abcd r 2 a 2 c 2 s 2 abcd acδ 1, so that rac + s abcd rac s abcd Thus th Lgndr symbol on th lt sid o 1.11 is indndnt o th choic o th intgr abcd modulo. Suos or th momnt that 1 mod. Din σ : sgn bc. Thnby1.5, abcd rac σs abcd sbdσ + r abcd δ Th irst symbol on th right sid o 1.13 can b valuatd using 1.11, in viw o I in addition to th hyothss o th thorm, w hav b, r c, s 1, 1.1 thn th scond symbol on th right sid o 1.13 can also b valuatd using 1.11, as ollows: in 1.11, rlac a by b ; b by a sgn b; c by d sgn c; d by c ; r by s; s by r; δ by δσ. Asarsult,whn1.1 holds 1 mod, our thorm can b alid to valuat th right sid o 1.13, thus roviding gnral critria or th quartic rsiduacity o intgrs having th orm abcd modulo. Examlso quartic rsiduacity critria ar givn throughout Sc. 3. I in addition to th hyothss o th thorm, w hav a b,

4 290 R. J. Evans & K. S. Williams thn ab cd may b takn as intgrs whos squars rsctivly qual ab cd modulo. In that cas, rom 1.11 th idntitis ra cd + sd ab c cd rac + s abcd, 1.15 rc ab + sb cd a ab rac + s abcd, 1.1 w can giv valuations o th symbols on th lt sids o , indndnt o th choic o squar roots o ab cd modulo. 2. Proo o Thorm From , w hav ab, cd,,, abcd From 1.5, , w dduc Not that by 1., , δt 2 ada 2 bcb B> Sinc 2,δ {±1, ±2}, abcdt, it ollows rom 2.2 that Nxt w show that From 1.10, w hav th two congruncs, A, B g : A, B rc sb mod g, sd ra mod g. 2. Multilying thm togthr, w obtain rscd 2 ab 2 0modg sothatby1.8, rs 0modg. Sinc g, 1by2., it ollows that g rs. By2., g bs 2 g ds 2,sothatby1.8, g s 2.Thusg s 2.Alsoby2., g cr 2 g ar 2, so that by 1.8, g r 2.Thusg r 2.Sincr 2,s 2 1by1., w hav g 1.This comlts th roo o 2.5. Now w show that h : B,t By 2.2, w hav h ada 2.Sinch, a divids both B a, it ollows rom 1.10 that h, a sd. Thus, by 1.8, h, a s. But a so h, a s. Sinca, s 1 by 1., w hav h, a 1.Hnch da 2,sothath da 2,B. But A, B 1by 2.5, so w hav h d. Ash B h d, whavrom1.10 thath ra. Sinc

5 Quartic rsiduacity th quadratic charactr 291 d, r 1by1.h d,whavh, r 1.Thush a. It ollows that h d, a, so that h 1by2.1. This rovs 2.7. From 1.8, w hav ± abcd cd mod. 2.8 By 1.12, , it ollows that rac + s abcd ±s abcd + rac c W rocd to valuat B. By , K 1, K 1, cd 2 ab 2, so that by alication o 1.2 to th rightmost symbol, w obtain cdk 1, B. 2.9 w cd 2, ab By 1.1, , B K 1, δk 1, ada 2 bcb 2, B B B so that by alication o 1.2 to th rightmost symbol, w obtain B a d B B δk 1, B w B ada 2, bcb Arguing in th sam ashion, w obtain a K 1,a sd w a sd, r 2.12 B B a By 1.1, d B a d d a K 1,d d B ra w d ra, s d d K 1,a d. 2.1 Th dsird rsult 1.11 now ollows by combining

6 292 R. J. Evans & K. S. Williams 3. Scial Cass o Thorm Th irst tn corollaris blow involv discriminants o binary quadratic orms or which thr is on orm class r orm gnus. For thos without rady accss to Pari or Sag, this can b chckd or 200 with th tabls in [1, ]. Thos tabls also indicat th orm classs. Mor xtnsiv tabls o orm class numbrs with may b ound in [7, ], th ormula in [13, Thorm 2.1] givs th sizs o corrsonding grous o gnra. For xaml, whn 20 as in Corollary 9, th orm class numbr is thr ar our gnra. Th signiicanc o on class r gnus is that, by Gauss s thory o gnra [5,. 221; 1,. 18; 7,. 17] th congrunc conditions or in our corollaris dtrmin th binary quadratic orm class rrsnting. Thr ar ininitly many xamls with on class r gnus that w could choos rom, bcaus thr xist ininit sts o ositiv discriminants with on class r gnus. Exlicit xamls o such sts ar givn in [, Sc. ]. It is conjcturd that thr xist ininitly many ositiv undamntal discriminants with on class r gnus. O cours this would b rovd i on could rov th xistnc o ininitly many ral quadratic ilds with class numbr 1. Th situation is dirnt or ngativ discriminants. Flath [7,. 198] lists 101 ngativ discriminants with on class r gnus, it is conjcturd that this list is comlt. For any nonzro intgr y modulo, wwrit y to dnot ithr o th two squar roots o y modulo. Asin[2,. 251], whn u is a squar modulo a rim 1modwithu, 1, w din { u 1 i u is a quartic rsidu modulo, 1 othrwis. I m is a ositiv squarr intgr, w writ ɛ m or th undamntal unit o th ral quadratic ild Q m. Our irst corollary is ssntially quivalnt to th classical rsult in algbraic numbr thory that an odd rim slits comltly in Q 2+ 2 Qcos2π/1 i only i ±1 mod 1 [1,. 27]. Corollary 13 will signiicantly xtnd Corollary 1. Corollary 1. Lt b a rim with 1 or 7mod8so that 2 is a squar modulo. Thn 2+ { 2 1 1/8 i 1 mod 8, /8 i 7 mod 8, ɛ2 1/8 i 1 mod 8, /8 i 7 mod 8.

7 Quartic rsiduacity th quadratic charactr 293 In th last quality it is undrstood that th choic o 2 modulo on th lt sid is th sam as that on th right sid. Proo. In 1.5, choos a 2 lt ach o th svn othr aramtrs qual 1. Lt b a rim such that 2 1, so that 1 or 7 mod holds. From th tabl in [1] or discriminant 8, thr ar ositiv intgrs such that 2 2 2, i holds. Clarly B +2 ar odd, { 0 mod 2 i 1 mod 8, 1 mod 2 i 7 mod 8. First suos that 1 mod 8, so that is vn. Th thorm givs w 2, 1 2 /2. Sinc 2 1 / / /8 1 1/8, w obtain 3.1 whn 1mod8. Now suos that 7 mod 8, so that is odd. In viw o 1., th thorm givs B w 2, 2 1 B 2 1 +/2 2. Sinc / /8 1 +1/8, w obtain 3.1 whn 7mod8. Finally, 3.2 ollows by multilying both sids o 3.1 by 2, sinc 2 2 whn 1mod8. W rmark that whn is a rim with 1 mod 8, it is a rsult going back to Gauss Dirichlt that D, whr C 2 +8D 2 ;sorxaml[10, Proosition 5.,. 15; 2,. 22]. Thn rom 3.2 w dduc ɛ2 1 D, which is a rsult o Barruc Cohn [1, 21].

8 29 R. J. Evans & K. S. Williams Thr ar two gnra o classs o orms o discriminant 2. Th rincial gnus contains th class o th orm x 2 y 2 th nonrincial gnus contains th class o th orm x 2 +y 2. Th nxt two corollaris involv th rrsntations o rims by ach o ths two orms x 2 y 2 x 2 y 2. Corollary 2. Lt b a rim congrunt to 1 or 19 modulo 2, so that is a squar modulo. Thn thr ar ositiv intgrs satisying such that / / / /8 i 1 mod 2, i 19 mod 2, i 1 mod 2, i 19 mod Proo. Choos a 2, b 3, c d r s t 1, δ 1. Thn 3.3 holds, in viw o th tabl in [1] or discriminant 2. From 3.3, w s that B +2 ar both odd, { 0 mod 2 i 1 mod 2, 1 mod 2 i 19 mod 2. I 1 mod 2, so that is vn, thn so that th thorm givs 2+ 1 B w 2, 1 /2, I 19 mod 2, so that is odd, thn 2 w 2, 1 / w 2, 1 +/2, so that th thorm givs w 2, /

9 Quartic rsiduacity th quadratic charactr 295 Suos irst that 1 mod 2. By 3.3, /8 1 /2 1 1/8, so that by 3., /8. This rovs 3. inthcas 1 mod 2. Th cas 19 mod 2 o 3. ollows rom 3.7 in a similar mannr. Finally, 3.5 ollows by multilying 3. by. Considr th scial cas o Corollary 2 whr 1 mod 2. W hav ɛ Hnc, by 3., ɛ 1 1/8 1 2, whn 1 mod 2, which is quivalnt to a rsult o Lonard Williams [11,. 103]. S 3.13 or anothr ormula or th lt sid o 3.5 whn 1 mod 2. Corollary 3. Lt 5b a rim congrunt to 5 modulo 2, so that is a squar modulo. Thn thr ar ositiv intgrs with odd odd satisying such that Proo. Choos a, b c 1, d 1, r 2, s 7, t 5, δ 1. Thn 3.8 holds, in viw o th tabl in [1] or discriminant 2. From 3.8 th congrunc 5mod8,wsthat, B ar odd. Sinc w 7, 2 1, th thorm givs B w 7, 2 1 B W hav 1 1 1, 3.11 B so w s that 3.9 ollows rom

10 29 R. J. Evans & K. S. Williams It ollows rom Corollary 3 that sinc 7+2 1, Corollary. Lt b a rim congrunt to 1 or 23 modulo 2, so that is a squar modulo. Thn or ach ɛ ±1, thr ar ositiv intgrs satisying ɛ Proo. Choos a, b ɛ, c ɛ, d r s t 1, δ 2ɛ. Thn 3.12 holds, in viw o th tabl in [1] or discriminant 9. From 3.12, w s that B + ar odd, ɛ mod 8. Th thorm givs ɛ +2 ɛ 2 w,. B Multilying both sids by ɛ, w obtain +2 2 B w, 2 B 3 w,. Sinc 1 B 1, it rmains to rov that 2 w,. B By 1., w hav /8 1 +3/2 w,, B which comlts th roo o Considr th scial cas o Corollary whr 1 mod 2. Thn by 3., ;

11 Quartic rsiduacity th quadratic charactr 297 multilying this by 3.13, w obtain th quartic rsiduacity critrion. For th roos o all rmaining corollaris xct Corollary 11, w omit th dtails, giving only th valus o th rlvant aramtrs in 1.5. Corollary 5. Lt b a rim congrunt to 1 modulo 2, so that is a squar modulo. Thn thr ar ositiv intgrs with odd vn satisying Proo. Choos a, b 1, c d s 1, r 2, t 5, δ 1. As 2 3 1, ,it ollows rom Corollary 5 that Corollary. Lt b a rim congrunt to ±1, ±9 or ±25 modulo 5, so that 7 is asquarmodulo. Thn or ach ɛ ±1, thr ar ositiv intgrs with odd vn satisying ɛ /2. Proo. Choos a 2, b 1ɛ, c ɛ, d s 1, r 3, t 2, δ ɛ. Considr th scial cas o Corollary whr ɛ 1,sothat is congrunt to 1,9or25modulo5.Whav ɛ7.

12 298 R. J. Evans & K. S. Williams Hnc, by Corollary, w obtain ɛ /2, whn 1, 9 or 25 mod 5, which is quivalnt to a rsult o Lonard Williams [11,. 103]. Corollary 7. Lt b a rim congrunt to ±1, ±9 or ±25 modulo 5, so that 1 is a squar modulo. Thn or ach ɛ ±1, thr ar ositiv intgrs with odd vn satisying ɛ Proo. Choos a 2, b 7ɛ, c ɛ, d s t 1, r 2, δ ɛ. For 1, 9 or 25 mod 5 w hav Hnc, by Corollary 7, w dduc ɛ1 2 2, whn 1, 9 or 25 mod 5, which is quivalnt to a rsult o Lonard Williams [11,. 103]. ɛ1 Corollary 8. Lt b a rim congrunt to 1 or 9 modulo 0, so that 10 is a squar modulo. Thn thr ar ositiv intgrs with odd satisying ɛ Proo. Choos a 2, b 20, c d s 1, r 3, t 1, δ 2. Corollary 9. Lt b a rim congrunt to ±1 or ±9 modulo 120, so that 15 is asquarmodulo. Thn or ach ɛ ±1 thr ar ositiv intgrs with

13 Quartic rsiduacity th quadratic charactr 299 odd vn satisying ɛ /2. Proo. Choos a 2, b 2ɛ, c 5ɛ, d 3, r s 1, t 2, δ ɛ. Considr th scial cas o Corollary 9 whr ɛ 1, so that is congrunt to 1 or 9 modulo 120. With a 2, b 2, c 3, d 5, r s 1, t 2, δ 1, th thorm givs /2. Togthr with Corollary 9, this yilds th quartic rsiduacity ormula Corollary 10. Lt b a rim congrunt to 1, 9, 25, 9 or 81 modulo 88, so that 22 is a squar modulo. Thn thr ar ositiv intgrs with odd odd satisying Proo. Choos a 2, b c 1, d 11, r s 1, t 3, δ 1. Each o our last our corollaris blow rovids an ininit class o xamls, on or ach q. Evry quadratic orm in ths corollaris rrsnts ininitly many rims in any arithmtic rogrssion comatibl with th orm s gnric charactrs, by a classical thorm o Myr [12]. S [15,. 72] or additional rrncs. Hnc, no xaml blow is vacuous.

14 2500 R. J. Evans & K. S. Williams Corollary 11. For any ixd odd intgr u, lt q : u 2 +2 so that q 3mod8. Lt b a rim congrunt to 1 modulo 8 with, qu 1such that q or ositiv odd intgrs. Thn 2+ 2q 2, 2q 2 q + 2q Proo. To rov th irst quality in 3.1, choos q, 3.1 q a 2, b c 1, d q, r s 1, t u, δ 1. Sinc q ar odd, so is B q +2. By1., 1 w 2 q,. Th thorm thus givs 2+ 2q 1 1 w 2 q,, 2 2 th irst quality in 3.1 ollows. For th scond quality in 3.1, choos a 1, b 2, c q, d r s 1, t u, δ 1. Thn B + is vn. By th thorm, th irst actor on th right sid o 1.11 quals q, ach o th rmaining ight actors quals 1. This rovs th scond quality in 3.1. Finally, 3.15 ollows rom th two qualitis in 3.1. Obsrv that Corollary 10 can b dducd rom th cas u 3 o Corollary 11. I u 1sothatq 3 1 mod 2, thn Corollary 11 shows that th unit 2+ 3satisis Corollary 12. For any ixd nonzro intgr u, lt q : u 2 +1 so that q 1 mod. Lt b a rim congrunt to 1 modulo with, qu 1such that q 2 2 or ositiv intgrs. Thn 1+ q 1, q + q 2 1, 3.1 q

15 Quartic rsiduacity th quadratic charactr 2501 q q Proo. To rov th irst quality in 3.1, choos a 2, b 2, d q, c r s 1, t 2u, δ 2. For th scond quality in 3.1, choos a 2, b 2, c q, d r s 1, t 2u, δ 2. Finally, 3.17 ollows rom th two qualitis in 3.1. Suos that th rim is congrunt to 1 or 9 modulo 20. Thn th hyothss o Corollary 12 hold with u 1q 5, by th tabl in [1] or discriminant 80. Thus by 3.1, ɛ This is quivalnt to a rsult o Lonard Williams [11,. 102]. By 3.17, A dirnt critrion or th quartic rsiduacity o 5 may b ound in [2,.217]. Combining , w s that ɛ which is a rsult o Lhmr [8, Eq. 11] i 1 mod 20, i 9 mod 20, Corollary 13. For any ixd odd intgr u, lt q : u 2 2so that q 7mod8. Lt b a rim with, qu 1such that ithr or q q or ositiv intgrs. In articular, ±1mod8. Thn q i 2, 2 1 /2 i

16 2502 R. J. Evans & K. S. Williams whn 1mod8, q q + 2q i 2, q 1 /2 i Thus whn 1mod8, w hav 2 q 2q 2q Proo. To rov 3.22 whn3.20 holds, choos i 2, i a 2, b 1, d q, c r s 1, t u, δ 1. To rov 3.22 whn3.21 holds, choos a 2, b q, c d r s 1, t u, δ 1. Now suos that 1mod8.Torov3.23 whn3.20 holds, choos a 1, b 2, c q, d r s 1, t u, δ 1. To rov 3.23 whn3.21 holds, choos a q, b 2, c d r s 1, t u, δ 1. Finally, 3.2 ollows by multilying togthr From th cas u 1o 3.22, w can dduc 3.1. Corollary 1. For any ixd odd intgr u, lt q : u 2 +1/2 so that q 1mod.Lt b a rim with, qu 1such that 2 2q 2 or ositiv intgrs so that 1 mod 8. Thn 2q 2q + 2q i 2, 2q 1 /2 i 2. Morovr, whn is vn, w hav 1+ 2q 1 1 /2, 2q 2q

17 Quartic rsiduacity th quadratic charactr 2503 Proo. To rov 3.25, choos a 2q, b c d r s 1, t u, δ 1. To rov 3.2 whn is vn, choos a 1, b 2q, c d r s 1, t u, δ 1. Rrncs [1] P. Barruc H. Cohn, Not on rims o ty x 2 +32y 2,classnumbr, rsiduacity, J. Rin Angw. Math [2] B. C. Brndt, R. J. Evans K. S. Williams, Gauss Jacobi Sums Wily- Intrscinc, Nw York, [3] H. Cohn, A Cours in Comutational Algbraic Numbr Thory Sringr, Brlin, [] H. Cohn, A numrical study o th rlativ class numbrs o ral quadratic intgral domains, Math. Com [5], Advancd Numbr Thory Dovr, Nw York, [] R. J. Evans, Rsiduacity o rims, Rocky Mountain J. Math [7] D. Flath, Introduction to Numbr Thory Wily, Nw York, [8] E. Lhmr, On th quadratic charactr o th Fibonacci root, Fibonacci Quart [9] F. Lmmrmyr, Rational quartic rcirocity II, Acta Arith [10], Rcirocity Laws Sringr, Brlin, [11] P. A. Lonard K. S. Williams, Th quadratic quartic charactr o crtain quadratic units I, Paciic J. Math [12] A. Myr, Übr inn Satz von Dirichlt, J. Rin Angw. Math [13] R. A. Mollin A. Srinivasan, Rsiduacity gnus thory o orms, J. Numbr Thory [1] W. Narkiwicz, Elmntary Analytic Thory o Algbraic Numbrs PWN, Warsaw, 197. [15], Th Dvlomnt o Prim Numbr Thory Sringr, Brlin, [1] H. E. Ros, A Cours in Numbr Thory Oxord Univrsity Prss, Oxord, 199. [17] Z.-H. Sun, On th quadratic charactr o quadratic units, J. Numbr Thory [18], Congruncs or A + A 2 + mb 2 1/2 b + a 2 + b 2 1/ mod, Acta Arith [19] M. Tsunkawa, Rlation ntr a+ m tlaloidsrésidus quadratiqus dans l cham d nombrs quadratiqus R m, Bull. Nagoya Inst. Tchnol English summary. [20] H. C. Williams, Th quadratic charactr o a crtain quadratic surd, Utilitas Math [21] K. S. Williams, Not on a rsult o Barruc Cohn, J. Rin Angw. Math [22] K. S. Williams, K. Hardy C. Frisn, On th valuation o th Lgndr symbol A+B m, Acta Arith