Proc. Amer. Math. Soc. 134(2006), no. 8, BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS Zhi-Wei Sun Department of Mathematics, Nanjing Universi

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1 Proc. Aer. Math. Soc , no. 8, 3. BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS Zh-We Sun Deartent of Matheatcs, Nanjng Unversty Nanjng 0093, Peole s Reublc of Chna zwsun@nju.eu.cn htt://web.nju.eu.cn/zwsun Abstract. Let E be a real quaratc fel wth scrnant 0 o where s an o re. For ρ ± we eterne 0<c<, c ρ c/ oulo n ters of a Lucas sequence, the funaental unt an the class nuber of E.. Introucton Let be an o re not vng a ostve nteger. A. Granvlle [G,.5] scovere the rearkable congruence 0<k< / + o, k/ where we use x to enote the ntegral art of a real nuber x. Subsequently the resent author [S] eterne further 0<k</ k/ o. In ths aer a ore sohstcate result connecte wth real quaratc fels wll be establshe. For A, B Z the Lucas sequences u n u n A, B an v n v n A, B n 0,,,... are gven by u 0 0, u, an u n+ Au n Bu n for n,, 3,..., v 0, v A, an v n+ Av n Bv n for n,, 3,.... It s well known that α βu n α n β n an v n α n + β n for every n 0,,,..., where α an β are the two roots of the equaton x Ax+B 0. Also, for any o re we have u o an v A o, where A 4B an 000 Matheatcs Subject Classfcatons. Prary B65; Seconary B37, B68, R. The author was suorte by the Natonal Scence Fun for Dstngushe Young Scholars No an the Key Progra of NSF No n Chna.

2 ZHI-WEI SUN enotes the Legenre sybol. See, e.g., [R,. 4-55]. If s an o re not vng B, then u snce Au + v u + an Au v Bu. Throughout ths aer, for an asserton P we set { f P hols, [P ] 0 otherwse.. Our an result s as follows. Theore.. Let E be a quaratc fel wth scrnant α r where α {0,, 3} an,..., r are stnct o res. Let ε a + b / be the funaental unt of the fel E where a, b Z, an Nε be the nor a b /4 of ε wth resect to the fel extenson E/Q. Let h be the class nuber of the fel E, an be an o re not vng. Then, for ρ ± we have 0<c< c ρ + ϕ α + [α > 0] + c/ 0< r + ρ [Nε] u a, Nεbh o,. where ϕ s Euler s totent functon an s the Kronecker sybol. Reark. Uner the contons of Theore., o 4 f α 0, an /4 3 o 4 f α ; also ves bu a, Nε snce for b we have a 4Nε b. Exale. Each of the quaratc fels Q 3, Q, Q 6, Q 7 has class nuber, an ther funaental unts are 3 + 3, 5 +, , wth nors,,, resectvely; see, e.g., [C,. 7]. Let be an o re an ρ {, }. If oes not ve 3,, 6, an 7, resectvely, then Theore.

3 gves the congruences 0<c<3 3 c ρ 0<c< c ρ 0<c<4 c, 6 c ρ 0<c<8 c, 7 c ρ BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS ρ 3 c/3 u 3 3,, ρ c/ ρ c/ ρ c/8 u 5,, 6 4u 6 0,, 7 4u 7 6, oulo resectvely, where 6 c an 7 c are Jacob sybols. We euce Theore. by cobnng the followng two theores. Theore.. Let > be an nteger wth the factorzaton α α r r where,..., r are stnct res an α,..., α r are ostve ntegers. Let be an o re not vng. Then ϕ + ϕ [r] k/ 0<k</ k, r α α + o..3 In the next theore we use the Bernoull olynoal B n x of egree n an the nth Bernoull nuber B n B n 0. Also, we let P enote the set of all ostve res. Theore.3. Let E be a real quaratc fel wth scrnant an class nuber h. Let ε a + b / > be the funaental unt of E where a, b Z, an Nε be the nor a b /4 of ε. Let be an o re not vng, an let u stan for bu a, Nε. Then c c B B c [Nε ] h u o,.4

4 4 ZHI-WEI SUN an 0<c</ c, c/ c { + hu o f 8 or P, + hu [Nε] o otherwse..5 Reark. In the case where o 4 s a re,.4 was rove n [GS] by eans of -ac logarths an Drchlet s class nuber forula see, e.g., [W]. In the srt of R. Cranall an C. Poerance [CP], Theores..3 ght be of coutatonal nterest. We shall ake soe rearatons n the next secton an gve roofs of Theores..3 n Secton 3.. On the su 0<k< k r k oulo Bernoull olynoals lay ortant roles n any asects. The reaer s referre to [IR,. 8-48] for basc roertes, an to [DSS] for a bblograhy of relate aers. In ths secton we rove the followng basc result an erve soe consequences. Theore.. Let be a ostve nteger not vsble by an o re. Then for any r Z we have k kr o k { r } { } r B B o,. where {x} stans for the fractonal art of a real nuber x. Proof. Alyng Lea 3. of [S3] wth k, we fn that j jr o j B + { } r { r } B o. For t {r /}, we have B + t l B t B l l + t t l 0 o. l Recall that B / an B n+ 0 for n,,.... Also, ves no enonators of B 0, B,..., B 3 by the theore of Clausen an von Staut cf. [IR, ]. Therefore. follows. Reark. The author frst scovere Theore. n Set. 99 by usng Fourer seres, an Lea 3. of [S3] was orgnally otvate by ths result.

5 BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS 5 Corollary.. Let an n be ostve ntegers, an let be an o re not vng. Then { n } B B n r n/ K r, k k k o,. where K r, : k k r k l l r l K r, o..3 Proof. In vew of Theore., n K r, r On the other han, n K r, r n r { r } { } r B B } B o. B { n n r k r k n r k j j, j n/ j k k k o. So we have.. Let be an o re an r be any nteger. An exlct congruence for K r, o aeare n Corollary 3.3 of [S]. By Theore. an [GS, 4] we can also eterne K 3 + 6r, 4, K 5, 40, K 5, 40, K 6, 60, K 36, 60 oulo n ters of soe secon-orer lnear recurrences. For a re an any a Z not vsble by, the Ferat quotent q a s efne as the nteger a /. Corollary.. Let be an o re an let be a ostve nteger not vsble by. Then we have rk r, q o..4 r

6 6 ZHI-WEI SUN Proof. By Corollary., n K r, n r n { n } B B { n } B B n r B B B o r0 where we have ale Raabe s theore n the last ste. B o cf. [IR,. 33]. Also, n K r, r K r, n r r So we have.4. r It s well known that + rk + r, sk s, o. Reark. It can be shown that.4 s equvalent to a forula of Lerch [L] whch was euce n a fferent way. s 3. Proofs of Theores..3 Proof of Theore.. For each ostve nteger we set ψ, c/ 0<c</ c, where ψ an ψ are consere as. For any a Z wth a, clearly a a a a a a / + a / a a a a / o. Thus, Theore. of [S] les that f 0 o then c/ 0<c</ { + f, + f, [ ] + [ ] o.

7 k/n n ψ for n,,..., alyng the Möbus nver- Snce 0<k<n/ son forula we get that ψ 0<c</ BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS 7 µ c/ µ/ [ ] µ/[ ] µ/ + µ [ ] By eleentary nuber theory, µ an also µ [ ] / µ [ ] µ 0 c c c snce >. Therefore ϕ ψ ϕ c / µ/ + µ o. o. Observe that µ/ / I µ I I {,...,r} r / I {,...,r} I r / r r r α r [r] r r. Also, ϕ + µ µ r r r r α α ϕ α ϕ + α r + α + α + α α α + α α + r α α α α α α + o.

8 8 ZHI-WEI SUN Thus.3 hols n vew of the above. Proof of Theore.3. Wrte ε V + U / where U, V Z, an let be an nteger wth o. Theore 3. of Wllas [W] states that h U Nε / β o where β 0<j<{ /} j. Let ε a b /. Then ε + ε a an ε ε Nε. Clearly v n a, Nε + u n a, Nεb ε n + ε n + εn ε n ε ε b ε n for n 0,,..., thus U bu a, Nε u an V v a, Nε. Observe that β j j j 0<< { /}>j j j<r< 0<< r j j j j<r< 0<< r K r,. j j<r< As χj j s a nontrval ultlcatve character oulo, the su j j vanshes. Therefore, wth the hel of Corollary., we have β j j K r, j r j j r 0 B { j j c K r, } + B { } j B B j c B B c o. Cobnng the above we obtan.4.

9 BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS 9 For each c,...,, we have χ c χ χc χc; also c/ c/ c/ 0<c</ c, k k + 0<c</ c, c/ k k B { c } B o. Takng the above congruence an.3 oulo, we obtan c/ c/ an hence 0<c</ ϕ c/ c [ s a re ower] o [8 or P]. Note that 4 ϕ an no square of an o re ves. On the other han, 0<c</ 0<c</ c/ c/ + 0<c</ 0<c</ c c B c B c c B r These, together wth.4, yel 0<c</ c/ c c { c } B { c } B B { c { c } B r B B r [8 or P] + hu } B o. [Nε] o.

10 0 ZHI-WEI SUN It s well known that Nε f 8 or P see, e.g., [C, ]. So the esre.5 follows. Proof of Theore.. By Theore. an the roof of Theore.3, [8 or P] + ϕ c/ F, o 0<c</ c, where F, [α > 0]α α + α + [α > 0] + 0< r + 0< r ; + also [8 or P] c + hu c/ 0<c</ c, where u bu a, Nε 0 o. Therefore 0<c</ c ρ + ϕ c/ + ϕ F, c/ F, + hu + ϕ F, + ρhu Ths roves.. We are one. 0<c</ c, [Nε] ρ + ρ hu [Nε] [Nε] o. [Nε] o +ρ c c/ Acknowlegent. The author thanks the referee for hs any helful coents. References [C] H. Cohn, Avance Nuber Theory, Dover Publ. Inc., New York, 96. [CP] R. Cranall an C. Poerance, Pre Nubers: A Coutatonal Persectve, Srnger, New York, 00.

11 BINOMIAL COEFFICIENTS AND QUADRATIC FIELDS [DSS] K. Dlcher, L. Skula an I. Sh. Slavutsk, Bernoull nubers, 73/990, Queen s Paers n Pure an Al. Math The webste of the on-lne verson s htt:// al.ca/ lcher/bernoull.htl. [G] A. Granvlle, Arthetc roertes of bnoal coeffcents.i. Bnoal coeffcents oulo re owers, n: Organc atheatcs Burnaby, BC, 995, 53 76, CMS Conf. Proc., 0, Aer. Math. Soc., Provence, RI, 997. [GS] A. Granvlle an Z. W. Sun, Values of Bernoull olynoals, Pacfc J. Math , [IR] K. Irelan an M. Rosen, A Classcal Introucton to Moern Nuber Theory Grauate texts n ath.; 84, n e., Srnger, New York, 990. [L] M. Lerch, Zur Theore es Feratschen Quotenten a / qa, Math. Ann , [R] P. Rbenbo, The Book of Pre Nuber Recors, Srnger, New York, 988. [S] Z. W. Sun, Proucts of bnoal coeffcents oulo, Acta Arth , [S] Z. W. Sun, On the su n kr o k an relate congruences, Israel J. Math. 8 00, [S3] Z. W. Sun, General congruences for Bernoull olynoals, Dscrete Math , [W] H. C. Wllas, Soe forulae concernng the funaental unt of a real quaratc fel, Dscrete Math. 9 99,

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