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Kummer system of congruences, the first case of Fermat's Last Theorem,. Stickelberger ... in the proof of nonvanishing of detP and detЯ (Theorem 3.3). Here, the.
transactions of the american mathematical society Volume 343, Number 2, June 1994

THE ORDERS OF SOLUTIONS OF THE KUMMER SYSTEM OF CONGRUENCES LADISLAVSKULA Abstract. A new method concerning solutions of the Kummer system of congruences {K) (modulo an odd prime /) is developed. This method is based on the notion of the Stickelberger ideal. By means of this method a new proof of Pollaczek's and Morishima's assertion on solutions of (K) of orders 3, 6 and 4 mod / is given. It is also shown that in case there is a solution of (K) ^ 0, ±1 (mod /), then for the index of irregularity i(/) of the prime / we have i(/) > [#7/2].

INTRODUCTION

In 1857, Kummer [12] introduced the following system of congruences (in equivalent form) for an odd prime /

(K)

Byfi-u = 0 (mod /)

( 1 \),

£>o= l.

These general indices of irregularity were introduced by Ernvall [8, 1985], where he also obtained an upper bound for them. For our needs we have had to improve Ernvall's bound, which was done by using the special matrices P

and Q in §3 (3.4, Theorem 4.5). In Pollaczek's other case, x has order 6 mod /, we use the inequality

x(Dl)(x)>(l-3)/2-iD(l)-p(l) (5.3), where p(l) means a special function defined in 4.7 with an upper bound

in 4.8. It seems that the depth of Pollaczek's and Morishima's Theorems consists in the proof of nonvanishing of detP and detß (Theorem 3.3). Here, the theorem \ja=i X(a)a ^ 0 for an odd Dirichlet character x of conductor / is used. At the conclusion we give a general theorem (7.6) on the existence of a nontrivial solution of the system (S). In this case we get i(/) > [\/T/2]. Note that Eichler [5, 1965] proved: "If the First Case of Fermat's Last Theorem for the prime / (/ > 3) fails, then lr (r — [v7] - 1) divides the first factor h~ of the /th cyclotomic field." It was observed by Bruckner [2] (and

Oberwolfach Conference 1975), Iwasawa, and Skula [19] ([27, §6.5]) that the statement lr/h~ can be substituted for i(/) > [VI] - 1 . Uehara [25] improved this result a little.

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SOLUTIONSOF THE KUMMER SYSTEMOF CONGRUENCES

589

1. Preliminaries Throughout this paper we will use the following notations:

/

an odd prime

r ind x

a primitive root mod / index of an integer x(l \ x) relative to the primitive root r

of I Z

the ring of integers ; the small latin letters will usually denote integers



the integer (/ € Z),

v

the integer with 1 < v < I - 1, v -v = I (mod /), where

0 < r, < I, r¡ = r' (mod /)

v eZ, l\v G

a multiplicative cyclic group of order / - 1

5

a generator of G, hence G = {I = s°, s, s2, ... , s'~2}

Ysfii = zC,=(A f°r suitable symbols¿, R = Z[G]

the group ring of the group G over the ring Z, thus

R~ =

R - {Y,ia¡s' '■ai e Z} ; f°r ;'êZ put a¡ - a, (0 the Stickelberger ideal of the ring R

I~ = R~~n /

the Stickelberger ideal of the ring R~

fa(t) = i(l) =

Y!~=Wávv?> e Z[t] for a = E/û/5' e R card{ 1 ')) = 0(mod/)

(l ldetÖI

Il ^(i2/c:

lfc=0 i:=0 w/jere C denotes a primitive I - 1st root of unity.

Proof. Put (as in the proof of 2.4) ( r, for r¡ < N y/(i) = \I l-n w for r¡>N

I

(0xy < I. Note that then the collections of sets {Xi, X2, ... , Xk} and {Yu Y2,...,Yk} are pairwise disjoint.

6.1.1. Proposition. The columns cy (ye YiU---llYk) of the matrix D¡(x) are linearly independent over the Galois field Z/IZ.

Proof. Let Y = Y\ U • • • U Yk and let ay eZ for each y e Y such that

Y,ayCy(yeY) = 0 (modi). y Then we have for each 1 < x < N J2ayxa^x'y)(ye

Y) + xY,ayxa{x>y](ye

Y) = 0

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(modi).

SOLUTIONSOF THE KUMMER SYSTEMOF CONGRUENCES

603

Denote by / the bijection from Y onto Y defined by

y = Yj ( 1 < j < k) =>f(y) e Yj, f(y) = -y (mod n). (This mapping is well defined because the sets Yj are complete sets of residues (mod n) and they are pairwise disjoint.) Since for each x e Xi and y e Y, a(x, y) = xy and (n - l)xy =

xf(y) (mod n), we have

0 = £^t^(^ e Y) + t£û>,t*/(% y y = y£^y(ay + Tany))(yeY) y

e Y) (mod /)

n-l

- £rXt £(^ (=0

+ tam)(ye

Y ' j' -? (mod ")) (mod o '

y

thus for each 0 < t < n - 1

(1)

£(ay + xafly))(y eY ,y = t (mod «)) = 0 (mod /). y Let Ac= 1. Then ay + T[^Y[ñ]. Proof. According to 6.1.1 and 6.2 we have r(D¡(x)) > kn , where Ac= [-l/2n v"«2 + 4/«/2«2] + 1.

+

The result follows by noting that Ac«> \Jl/n - 1 . Using Proposition

6.4.

6.3 and Theorem 2.3 we get

Theorem. Let 5 < « < ^ 1/2 and let x be a solution of the system (S) of

order « mod /. Then i(l) > [\JTjn]. 1. Solutions

of (S) with "large"

orders

and conclusion

7.1. Proposition. Let 1 < m < i///2 and let x be an integer of order mod / greater than 2m . Then r(D/(x)) > m . Proof. We show that the first m columns Ci, ... , cm of the matrix D¡(x) are linearly independent over the Galois field Z//Z. Let ay e Z (l < y < m) such that £ 2, hence / > 157. Proof. According to Theorems 5.3 and 5.5 we have for the order « of x mod /,

« > 5 and « # 6, hence / > 11 . According to 6.4 and 7.4 we have

i(/) > min| yfl/S , [v^7]| > 1 and therefore i(/) > 2 for / > 32 . The result follows from the tables of irregular primes [1, Table 9]. 7.6. Theorem. If there exists a solution x of the system (S), x ^ 0 (mod /),

t =á-1 (mod /), then i(l) > [f/Tjl]. Proof. We can assume according to 5.3 and 5.5 that « > 5, where « is order of x mod /. Further, according to 7.5 assume I > 157.

It followsfrom 7.4 i(/) > [-JÏJQ> [i/TJl] for « > y/TJl. If n < \Í4l, then « < y/T/2 and according to 6.4 i(/) > [y/T/n] > [{/T/2]. If \[4l < « < y/T/2 put m = [(« - l)/2]. Then we get from 7.2 i(/)>[(H-l)/2]>^-l>^///2-l,

hence i(l) > [1/7J2]. 1.1. Remark. Under assumptions of 7.6 we have / > 150,000 and i(/) > 42. Proof. By the statement in Remark 7.5, / > 157, so according to Theorem 7.6, i(/) > 4. Using again the tables of irregular primes (e.g. [1, Table 9]) we get

/ > 432, therefore by Theorem 7.6, i(/) > 6. Since, according to the latest tables for the index of irregularity by Tanner

and Wagstaff[24, 1987] and Wagstaff [26, 1978], i(/) < 5 for /< 150,000, Theorem 7.6 concludes the proof.

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606

LADISLAVSKULA

Using quite recently made out tables of the index of irregularities by Buhler,

Crendall and Sompolski [3, 1991] (maxi(/) = 6 for / < 106) we get / > 106 and

i(/) > 62.

7.8. Remark. By Washington [27, Chapter 6, §6.5] probability arguments indicate i(/) = 0(log//loglog/). Therefore we can conclude (1.8, 1.9(a), (c) and 7.6) that besides x = 0 (mod /) and x = ±1 (mod /) the Kummer system of congruences has with great probability no solution.

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of Science, Masaryk

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University,

Janáckovo