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18Вестник Сыктывкарского университета. С ер Л. Вып. 15.2012
УДК 511.92
ANKENY, ARTIN AND CHOWLA CONJECTURE FOR EVEN GENERATORS
A. Grytczuk
In the paper [2] Ankeny, Artin and Chowla conjectured that if p is a prime such that p=l( mod 4) and 6 = t+u^/P js fundamental unit of the real quadratic field К — Q (у/p) then p \ u.
In this paper we prove that this conjecture is true for even generators £, и of the fundamental unit e.
Primary subjects: 11 A 55, 11D 09 Secondary subjects: 11 R 11 Keywords: генераторы, простые числа, диофантовы уравнения
Introduction. Let К — Q (у/p) be the quadratic number field with the prime number p = 1 ( mod 4). Then it is well-known that the fundamental unit e has the following form:
(1.1) £=*+%&> 1,
where the generators t, и are the same parity. Moreover, it is known that the norm N (e) of this unity is equal to — l,so N (s) = — 1. From this fact immediately follows that
(1.2) t2 — pu2 = —4,
From (1.2) follows that for the solution of the Ankeny, Artin and Chowla conjecture it suffices to consider the Diophantine equation (1.2).
We remember that Mordell [7] proved that if prime p = 5 ( mod 8) then и = 0 ( mod p) if and only ifg the Bernoulli number Bp^i = 0 ( mod p).
4
This criterion for remaining primes p = 1 ( mod 4) has been proved by Ankeny and Chowla in the paper [3].
@ Grytczuk A., 2012.
Moreover, in the paper [4] Chowla remarked some intersting congruence relation connected with the class number h of the field K = Q (^fp) , p = 1 ( mod 4) .Namely, we have
(1.3) (2^)! = (-1)^ • §( modp).
Another criteria cinnected with A AC conjecture has been given by Agoh [1] and by Yokoi [10]. In the paper [8],[9] Sheighorn obtained interesting connections between fundamental solution (x0,y0) of the negative Pell’s equation
(1.4) x2 — py2 = —1
with p = 1 ( mod 4) and the manner of reflection lines on the modular surface and the ^fp Riemann surface.
In the paper [5] has been given two new criteria connected with AAC conjecture. In this purpose has been used the representation of ^Jp as a simple continued fraction.
It is easy to see that if the generators t, u are even, so t — 2x0, u — 2y0 the the equation (1.2) reduce to the equation
(1.5) xl~pyl = ~ 1.
In this paper we prove that AAC conjecure is true for even generators t = 2^o, u = 2yo-Namely we prove of the following theorem:
Theorem. If p is a prime number such that p = 1 ( mod 4) and (xq, i/o) is the fundamenal solution of the equqtio (1.5) then p\u.
In the proof of the Theorem we use Lemma 1 and Lemma 2, whoes been proved in our paper [5] as the Theoreml,Theorem 2 and Lemma 3.
2. Basic Lemmas
Lemma 1. Let p be a prime number such that p = 1 ( mod 4) and let p = b2 + c2,(b, c) = 1. Moreover, let ^/p = [çoî Qi, Ç2, •••, Çs] be the representation of y/p as the simple continued fraction and let (xo,yo) be the fundamental solution of the equation (1.5) . Then p \ y0 if and only if
(2.1) p | cQr + bQr_i and p \ bQr — cQr_i,
where r = ^ and 7T is the n — th convergent of the simple continued
£ Wn
fraction of yfp.
Lemma 2, Let be satisfied of the assumption of the Lemma L Then p | 2/0 if and only if
(2.2) p | 4&QrQr_1 - (-l)r+1.
Moreover, we have
(2.3) iVi — PrQr ± Pr-iQr-i-, Qs-i = Of- ± Qi-1-
3.Proof of the Theorem.
Suppose that p | ^ Eor further consideration we use of the following well-known properties of the divisibility relation:
(Ri) if d | a and k ^ 0 , then d \ k • u ,
(R2) if d | a and d \ F, then d | a + b and d \ a — F,
where d, a, b, k are integer numbers.
From (Ri) and (2.1) it follows that p | 4cQ2 + 4bQrQr_i. Hence, from (2.2) and the relation (R2) we get
(3.1) p | 4cQ2r + (-l)r+1.
In similar way from the second relation of (2.1), relation (2.2) and (R2) we obtain,
(3.2) p\4cQ2r_1-(-l)r+1.
By (2.2) of Lemma 2 and (Ri) it follows that
(3.3) p | 4bcQrQr-i — c (—l)r+1.
On the other hand from (3.2) and (Ri) we have
(3.4) p\4bcQ2r_1-b(-l)r+1.
92 Grytczuk A.
From (3.4),(3.3) and (R2) we obtain
(3.5) p | 4bcQr_i (Qr + Qr—i) — (—1) (b + c).
By completely similar way it follows that
(3.6) p | 4bcQ2 + b(-l)r+1, and
(3.7) p | 4bcQr (Qr — Qr-1) + (—l)r+ (b + c).
From (3.5) and the relation (Rx) with k = Qr — Qr-i we obtain
(3.8) p | 46cQr_! - Ql_x) - (-l)r+1 (b + c) (Qr - Qr-1) •
In similar way from the relation (Ri) with k = Qr + Qr-i and (3.7) we
get
(3.9) p | 4bcQr (Qr ~ Qr-1) “I- (—1) (^ “I- c) (Qr Qr—i) •
By (3.8),(3.9) and the relation (R2) it follows that
(3.10) p I 4be (Ql - Ql_ J (Qr - Qr-i) + 2 (-l)r+1 gr_! (b + c).
We known that fundamental solution of the equation (1.5) is given by the formulas:
(F) Xq Ps—i, yo Qs—i-
From (2.3) of Lemma 2 we have that Qs-± = Q2r + Q2_ 1 ■ Hence, by the assumption that p \ y0 and second formula of (F) it follows that
(3.11) p | Q2 + Qr~i-
It is easy to see that the following identity is true:
(3.12) (Q2 — Q2_i) (Qr — Qr-i) = (Qr — Qr-1)2 (Qr + Qr-1) =
[(Qr + Qr-1) — 2QrQr-i] (Qr + Qr-1) •
From (3.12),(3.11),(3.10) and (3.1) we obtain
(3.13) p | —8bcQrQr-i (Qr + Qr—i) “I- 2 (—1) Qr—i (b + c).
By (2.2) of Lemma 2 we have that p \ Qr~i and consequently from well-known property of the divisibility relation and (3.13) we obtain
(3.14) p I 4bcQr (Qr + Qr-1) — (—1) (b + c).
From (3.14) and (3.7) we get
(3.15) p I 4bcQr (Qr + Qr-1 + Qr ~ Qr-i) •
The relation (3.15) implies that
(3.16) p I AbcQ2.
We observe that the relation (3.16) is impossible.In fact by (2.2) of Lemma 2 it follows that p \ Qr and consequently p \ Q2. Since p — b2 + c2 and (b, с) — 1 then we have that p\b and p \ c.
Hence, we obtain a contradiction and the proof of the Theorem is complete.■
Литература
1. T.Agoh, A note of unit and class number of real quadratic fields, Acta Math.Sinica 5(1989),281-289
2. N.C.Ankeny,E.Artin and S.Chowla,The class number of real quadratic fields, Annals of Math.51(1952),479-493
3. N.C.Ankeny and S.Chowla, A note on the class number of real quadratic fields,Acta Arith.6(1960),145-147
4. S.Chowla,On the class number of real quadratic fields, Proc.Nat. Acad.Sci.U.S. A.47(1961),878
5. A.Grytczuk,Remark on Ankeny,Artin and Chowla conjecture,Acta Acad.Paed.Agriensis,Sectio Mat.24(1997),23-28.
6. R.Hashimito,Ankeny-Artin-Chowla conjecture and continued fraction, J. N umber Theory, 90 (2001 ) ,143-153.
7. L.J.Mordell,On a Perllian equation conjecture,Acta Arith.6(1960),137-144.
8. H.Sheigorn, Hyperbolic reflections on Pell’s equation,J.Number Theory 33(1989),267-285.
9. H.Sheigorn,The у/p Riemann surface,Acta Arith.63(1993),255-266.
10. Y.Yokoi,The fundamental unit and bounds for class numbers of real quadratic fields,Nagoya Math.J.124(1991),181-197.
Summary
Grytczuk A. Ankeny, Artin and Chowla conjecture for even generators Keywords: generators, prime numbers, diafantic equation.
Department of Mathematics and Applications Jan Pawel II Western Higher School
of Marketing and International Finances Поступила 27.04-2012
