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JOURNAL OF NUMBER THEQRY 26, 325-367 (1987)
Solving Exponential Diophantine Equations Using Lattice Basis Reduction Algorithms
B. M. M. DE WEGER*
Mathematisch Insfiluuz, Rijks Uniuersireit Leiden, Postbus 9512, 2300 RA Leiden, The Nether1and.s
Communicated by M. Waldschmidt
Received June 5, 1986
Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0 < x - y < y” in x, y E S for Iixed 6 E (0, 1 ), and for the diophantine equation x + y = z in x, y, 2 G S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L’-Basis Reduction Algorithm. Elaborate examples are presented. el 1987 Academic
Press. Inc
1. INTRODUCTION
In 1981, L. LovQsz invented an algorithm for computing a reduced (i.e., nearly orthogonal) basis of an arbitrary lattice in [w” from a known basis of the lattice. It has a surprisingly good theoretical complexity (polynomial time), and also performs very well in practice. This algorithm, together with an application to the factorization of polynomials, is described in Lenstra et al. [9]. It has several other interesting applications, such as in public-key cryptography (cf. Lagarias and Odlyzko [8]), and in the dis- proof of the Mertens conjecture (cf. Odlyzko and te Riele [13]). We shall refer to the algorithm as the “L3-Basis Reduction Algorithm,” (L3-BRA).
The L3-BRA can also be used for solving multi-dimensional diophantine approximation problems, as Lenstra et al. already indicated [9, p. 5251. In the present paper it is shown that this enables us to find all solutions of certain exponential diophantine equations and inequalities in a routine manner. As is well known, many types of diophantine problems are associated to linear forms in logarithms of algebraic numbers (see, e.g., Baker [3, Chap. 41, Shorey and Tijdeman [ 181, Stroeker and Tijdeman [20, pp. 343-3531). Namely, for any large solution of the diophantine problem some linear form in logarithms is extremely close to zero. The Gelfond-Baker method provides effectively computable (and in many cases
* Present address: Department of Applied Mathematics, University of Twente, P.0. Box 217, 7500 AE Enschede, The Netherlands.
325 0022-314X/87 $3.00
CopyrIght ,(‘ 1987 by Academx Press, Inc All nghta of reproductmn m any form reserved
326 B. M. M. DE WEGER
explicitly computed) lower bounds for the absolute values of such linear forms. Thus, explicit upper bounds for the solutions of many diophantine problems can be obtained. The bounds that are found in this way are so large that enumeration of the remaining possibilities is practically impossible. However, it is generally assumed that the bounds are far from the actual largest solution. Therefore it is worthwile to search for methods to reduce the found upper bounds.
If the linear form in logarithms under consideration has only two terms, a simple method applies, based on continued fractions. For example, Cijsouw, Korlaar, and Tijdeman (Appendix to Stroeker and Tijdeman [20]) found in this way all solutions of the diophantine inequality
1 p v - q I 1 < p”-x- (1.1)
for all primes p, q with p < q < 20, and 6 = 4. In Section 4.B we extend this result for many more values of p, q, and 6 = 6.
A natural generalization of (1.1) is the following problem. Let S be the set of all positive integers composed of primes from a fixed finite set { pI ,..., p,}, where t > 2, and let 6 E (0, 1). Then lind all solutions of the diophantine inequality
0 < .Y - 1’ < y’ (1.2)
in x, y E S. Putting X/-V = p;’ . ‘p:‘, the corresponding linear form in logarithms is
/i=.u,logp,+ ..’ +x,logp,.
The continued fraction method applies only for t = 2. For f 2 3, multi- dimensional continued fraction algorithms are available (cf. Brentjes [S]), but they are not useful for our purposes. In Section 4.C we shall show that the L3-BRA leads to substantial improvements of the upper bounds for (1.2). Usually the new bound is of the size of the logarithm of the initial bound. For t = 6, {p, ,..., p6} = { 2, 3, 5, 7, 11, 13 >, 6 = 4, we show in detail how (1.2) can be solved completely with this method.
If the linear form is inhomogeneous of the form
/l=x,logcr, + .‘. +x,loga,+loga,+,,
it can of course be made homogeneous by introducing the variable x,+ i as coefficient of the last term. We may then solve this (n + 1)-dimensional approximation problem, and select all solutions with x,+ , = 1. There is, however, a different approach, which may ,be faster. See Baker and Daven- port [4] for n = 2, and Ellison [6 3 for n > 2. It is then needed to find good simultaneous approximations pi/q to log a,/log a, (i = l,..., n - 1). Lenstra
DIOPHANTINE EQUATIONS 321
et al. [9, p. 5251 have indicated how the L3-BRA can be used to find such approximations. We do not work this out in the present paper.
Up to now we have only considered real linear forms in logarithms. There is a p-adic counterpart of the Gelfond-Baker theory, which provides lower bounds for the p-adic values of linear forms in p-adic logarithms of algebraic numbers. It is therefore a natural problem to devise p-adic analogues of the diophantine approximation methods sketched above. The simplest case is that of an inhomogeneous form with only one variable, such as
A =x log, a, + log, c(2.
Then it suffices to compute the p-adic expansion of log, aJog, a, far enough. See Wagstaff [21], Petho and de Weger [14], and de Weger [24].
In the case of a form with two terms, such as
/1=x,log,a,+x,log,cr,
a practical p-adic analogue of the real continued fraction algorithm is needed. Such an algorithm was first formulated by Mahler [ 11, Chap. 41. A similar algorithm has been studied by de Weger [23] in the context of p-adic approximation lattices. See Agrawal et al. [ 1 ] for an application to a Thue-Mahler equation. We shall show in Section 5.C how to solve
p-y + pi? = )$lp’;’ (1.3
for fixed pr , p2, pj, w using this algorithm. A natural generalization of (1.3 is the diophantine equation
x+y=z (1.4)
in x, y, z E S, with S as above. We may assume gcd(x, y, z) = 1. Put p = pr, and suppose p 1 z. Then p i xy. Put x/y = p-T’ *. .pf?j. Then we have the p-adic linear form in logarithms
n =x1 log/$, + ... S-X,-I 10$&p,-,.
The concept of approximation lattices of p-adic numbers, as introduced in [23], can be generalized to the multi-dimensional case, as we shall see in Section 5.B. Then we can apply the L3-BRA. In Section 5.D we show how this can be used to solve (1.4) explicitly. We give details for t = 6, {pl ,..., p6} = (2, 3, 5, 7, 11, 13). This generalizes the results of Alex [2], who gave a complete solution of (1.4) for t = 4, {pl ,..., p4} = (2, 3, 5, 7) by elementary arguments. The case where z has only one prime divisor was treated by Rumsey and Posner [16], also by elementary means.
Many diophantine equations, such as the Thue equation, the Thue-
328 B.M.M.DE WISER
Mahler equation, the hyperelliptic equation and the Mordell equation, lead to linear forms in logarithms similar to those described above. These equations differ from our examples (1.2) and (1.4) in that the path from the equation to the linear form in logarithms is not as straightforward; it leads through some algebraic number theory. This clearly does not affect the applicability of our approximation methods for reducing upper bounds, since they are based only on the linear forms themselves.
2. BOUNDS FOR LINEAR FORMS IN LOGARITHMS
In this section we quote the results that we use from the theory of linear forms in logarithms. We do not quote the theorems in full generality, since we apply them only for logarithms of rational integers, and for rational coefficients. The results provide lower bounds for linear forms in logarithms in the real and p-adic cases. We chose results that give completely explicit constants and lead to convenient upper bounds for the solutions of the diophantine problems we want to solve. We stress that our methods for reducing these bounds are in principle independent of the size of the bounds.
Let p, ,..., pI1 (n > 2) be rational integers such that 2 <p, < .. <p,,, and [Q(p;l’,..., pf,“): Q] = 2”. Let b ,,..., h,,eZ’, and put B=max, SiG,l IhJ. In the real case we have the following result.
LEMMA 2.1. (Waldschmidt). Let
n =h, logp, + .” +h,,logp,,
he nonzero. Put
V, = max( 1, logpi) (i= l)..., n), Q= V, ... V,,,
c,=2 “I + 26n’7 + 4!2 log( e V,, , ), c, = c, log(e v,, ).
Then
IAl >exp(-(C, log B+C,)j..
This lemma was proved by Waldschmidt [22]. In the case n = 2 a sharper bound was given by Mignotte and Waldschmidt [ 121. In the p-adic case we have the following result:
LEMMA 2.2. (van der Poorten). Let p be a prime with p i pi (i = l,..., n). Let
A =h, log,p, + ... +h,log,p,,
DIOPHANTINE EQUATIONS 329
be nonzero. Choose p, ti with 2/(n + 1) d p 6 2, 0 < K < p/2. Put
Vi = max(e, log pi) (i= l,..., n), Q= If,*.. v,,
G = P(P-1) P
b
if p=2,3
-1 if p>,5,
&=@--)/(I +K)(l +p)(n+l),
~=max{(~6n)‘~+““““+“, (@)(‘+V’)(“+‘), 161’“},
C,=4(n+ 1) cn+“k(l+~‘l(Gp/logp) Q.
Then B < I, OY
ord,( /i ) < C,(log B)‘.
This lemma follows from the proof of Theorem 2 of van der Poorten [IS]. Note that we omitted the factor n(2D2)“+ I, since D = 1; cf. van der Poor- ten [ 15, p. 351. To save computation time we may choose p, K as a function of n such that C3 is minimal, for van der Poorten’s estimate (16(n + l))“(“+‘) for 4(n + l)(ncl’k(‘+V) (with p = 2, K = +) is rather crude. Note that for n = 2 a sharper bound was by Schinzel [ 171. It is expected that the constant C3 of Lemma 2.2 can be sharpened considerably (van der Poorten, private communication).*
We also need the following simple lemma. For its proof, see Petho and de Weger [14, Lemma 2.23.
LEMMA 2.3. Let a> 0, h > 1, b > (e’/h)“, and let x E R satisfv .Y 6 a + b(log x)“. Then
x < (2a”” + 2b’lh log(hhb))h.
3. THE L3-B~srs REDUCTION ALGORITHM
In this section we describe how we use the L3-BRA. All lattices that appear in this paper are integral lattices, that is, sublattices of Z”. In the algorithm as stated in [9, Fig. 1, p. 5211, non-integral rationals may occur, even if the input consists of rational integers only. We now describe a variant of the L3-BRA in which only integers occur. This has the advantage of avoiding rounding-off errors.
Let Tc Z” be a lattice with basis vectors b, ,..., b,. Define b:, pu, di as in [9], (1.2), (1.3), (1.24), respectively. The di can be used as denominators
* Note added in proof. Recently, K. R. Yu has obtained such an improvement, to be published in the Proceedings of the 1986 Durham Conference. His results lead (in Section 5) to bounds less than the square root of the bounds we derived using Lemma 2.2.
330 B. M. M. DE WEGER
for all numbers that appear in the original algorithm [9, p. 523). Thus, put for all relevant i, j,
(3.1 1
(3.2)
They are integral, by [9], (1.2X), (1.29). Note that, with Bi= lb,Fj2.
d, = di , Bi. (3.3)
We can now rewrite the L3-BRA in terms of ci, d,, i.,,j instead of b,?, Bi,
pi.,, thus eliminating all non-integral rationals. We give this variant of the algorithm in Table 1. All the lines in this variant are evident from applying
TABLE I
Variant of the LA-Basis Reduction Algorithm
(A)
d,:= I: c,:= b. A., : = ;i,, c,); c, := (d/c,-A,,c,)/d,-,
fOfj= I,..., i- 1; for i== I, .__, n;
4:= (%C;)ld,. , k:= 2; /
(1) perform (*) for /=&I; if 4dke2dk <3a;-, -4;1;,,_ ,, go to (2); perform (*) for I = k - 2,.... 1; if k = n, terminate; k:= k+I;
go to ( 1 L
(Cl 4-l := (dk-zdk+~:.lr-,)/d~.l; ifkz2, then k:= k-l;
go to (1); (*) if 2 I&,,( > d,, then
r := integer nearest to &Jd,; b, := be-rb,; uk := u,-rru,; A,, := &,-rA,,, for j= l,..., I- 1; A,,, : = &, - rd,.
DIOPHANTINE EQUATIONS 331
(3.1), (3.2), and (3.3) to the corresponding lines in the original algorithm, except the lines (A), (B), and (C), which will be explained below.
We added a few lines to the algorithm, in order to compute the matrix of the transformation from the initial to the reduced basis. Let C be the matrix with column vectors b,,..., b,, (we say: the matrix associated to the basis b, ,..., b,), and let B be the matrix associated to the reduced basis computed by the algorithm. Then we define this transformation matrix Y by B = CV. More generally, let U be the matrix of a transformation from some C, to C, so C = C,,U. Denote the column vectors of U by u, ,..., II,,. All manipulations in the algorithm done on b, ,..., b, are also done on UI ,**-, u,. Then the algorithm gives as output matrices B and u’, such that B is associated to a reduced basis, B = CV, and u’ = UV. (Note that V is not computed explicitly). Hence B = CU-’ U’ = Co U’, so u’ is the matrix of the transformation from C, to B. In particular, if U = I, then C = Co and U’= v.
We now explain lines (A), (B), and (C).
(A) From [IS], (1.2) it follows that
Define for j=O, l,.,., i- 1,
ci(j)=djbi- .f $f& &k~k. k=l k-1 k
Then c,(O) = bi, and cj( i - 1) = ci. The ci( j) is exactly the vector computed in (A) at the jth step, since
tdjc,(j- l)FA,jcJ)ldj- I J-1
zdjb,- C A AikCk-A n;,.jCj=ci(j).
k=l dk-,dk ’ J-1 I
This explains the recursive formula in line (A). It remains to show that the occurring vectors q(j) are integral. This follows from
d, i -!- k=l h-,4
Ai,kCk=dj i P;,kbk*, k=l
which is integral by [9, p. 523, 1. 111.
(B), (C) Note that the third and fourth line, starting from label (2), in the original algorithm are independent of the first, second, and fifth line. Thus a permutation of these lines is allowed. We rewrite the first, second,
641126/3-l
332 13. M. M. DE WEGER
and fifth line as follows, where we indicate variables that have been changed by a prime,
B; - 1 : = B, + ,L& ~, B, , ; (3.4)
B;:= B,m,B,/B;p,; (3.5)
/L;,~-, := p+,Bk-,IB;. -1: (3.6)
i4.k - I := &k-l~,.k-l+(l -pk.k I&k~I)/h,k;
1
(3.7) for i = k + l,..., n.
p1.k : = pLi,k - I - pk,k I pi.k (3.8)
The d, remain unchanged for i = 0, l,..., k - 2, and by (3.5) also for i = k. Now, (3.4) is equivalent to
d;-, dk A:,-, h-1 d, >=clfi+K d,mz’
(3.9)
which explains (C). From (3.6) we find .I lb,, I j.h.k I 4 , 4 2 A==--- 4 I 4 I dk 7 d;. , ’
hence &.& , remains unchanged. From (3.7) we obtain
whence, by multiplying by d,
dk , 3.i.k - , = &.A-
= &,k
Finally, from (3.8) we see
~, d;, _ , and using (3.9)
and (B) follows. In our applications we often have a lattice r, of which a basis is given
such that the associated matrix, A say, has the special form 1 0
Acoy , t 1 8, ... 8,
DIOPHANTINE EQUATIONS 333
where the 13, are large integers (they may have several hundreds of digits). We can compute a reduced basis of this lattice directly, using the matrix A itself as input for the L3-BRA. But it may save time and space to split up the computation into several steps with increasing accuracy, as follows.
Let k be a natural number (the number of steps), and let 1 be a natural number such that the fIi have about kl (decimal) digits. For i= l,..., n, j = I,..., k, put
and define $I” by
fl(.i+l)= 1O’Q!jJ+$)jJ. I I
Thus, the +I” are blocks of 1 consecutive digits of Oi. Define for the relevant j,
i
1
E= 0’
Then it follows at once that
Ai+, = EA, + Yj.
Note that A, = A, since elk’ = Bi. Put UO = Z, C, = A,. For some j 2 1 let C, and U,- , be known matrices. Then we apply the L3-BRA to C= C,, Ii = U, _, . We thus find matrices B, and U, such that
Now put
Bj= CjU,:‘, U,.
c I+1 = EBj+ !P,Uj.
By induction the matrices B,, C,, and U, are well defined for j= l,..., k. Note that
334 R. M. M. DE WEGER
so the C,U, I, satisfy the same recursive relation as the A,. Since C, U;’ = A,, we have C,U, !, = A, for all j. Hence
Bi= C,lJmml, U,= A,U,.
and it follows that B, and A, are associated to bases of the same lattice, which is ZY Moreover, since B, is output of the L3-BRA. it is associated to a reduced basis of f.
Let us now analyse the computation time. For a matrix A4 we denote by L(M) the maximal number of (decimal) digits of its entries. If the L3-BRA is applied to a matrix C, with as output a matrix B, then according to the experiences of Lenstra, Odlyzko (cf. Lenstra [ 10, p. 71) and ourselves, the computation time is proportional to L(C)3 in practice. Since B is associated to a reduced basis, we have
L(B) ‘v ‘Olog(det(r))/n.
In our situation, L(A,) 2: fj, L( Y,) N 1, and since det(Bj) = det(A,) = Bi”, we have L(B,) 2: lj/n. Put B,= (b$j’), U, = (u:,i’). Then by B,= A,U, and the special shape of A, we have b,,, ( j) = ~fjh) for i = l,..., n - 1, h = I,..., n, and
It follows that L( U,) N L(B,). So
Instead of applying the L3-BRA once with A as input, we apply it k times, with Cl,..., C, as input. Thus we reduce the computation time by a factor
For k between 2.5n and 3n this expression is maximal, about 0.4n2. So the reduction in computation time is considerable. The storage space that is required is also reduced, since the largest numbers that appear in the input have I( 1 + ((k - 1)/n)) digits.
We use the L3-BRA for finding a lower bound for the length of the non- zero vectors of a lattice f. Let 1.1 denote the euclidean length on [w”. Put
Then the following inequality holds (cf. [9, (l.ll)]).
DIOPHANTINE EQUATIONS 335
LEMMA 3.1. (Lenstra et al.). Let b, ,..., b, be a reduced basis of the lattice IY Then
l(f)>,2- (n-l)/2 (b,l,
In some applications we want to compute all vectors in a lattice with length bounded by a given constant. To do this we employ a recent algorithm of Fincke and Pohst [7], in combination with the L3-BRA.
4. A DIOPHANTINE INEQUALITY
Let p, < . . . <pr be prime numbers, where t > 2. Let S be the set of all positive integers composed of these primes only, so
S= {p;1...p.$x,~Z, x,20 for i= l,..., t}.
Let 0 < 6 < 1 be a fixed real number. We study the diophantine inequality
O<x-y<yb (4.1)
in x, y E S. For a solution x, y of (4.1), the finitely many z E N for which zx, zy is also a solution of (4.1) can be found without any difficulty. Therefore we may assume that (x, y) = 1. Put
Tijdeman showed that there exists a computable number c, depending on p, only, such that for all x, y E S with x > y 2 3,
x - y ’ Yl(h2 Y 1’
(cf. Shorey and Tijdeman [ 18, Theorem 1.11). Thus, for any solution of (4.1) a bound for X can be computed, and we do so in Section 4.A. In Sections 4.B and 4.C we show how to reduce such an upper bound, in the cases t = 2 and t 2 3 respectively.
4.A. Upper Bounds
THEOREM 4.1. In the above notation, put
C 4- -29”“t’14max(1, &) logp,~~~logp,log(elogp,~,)/(1-6),
C5 = 2 log 2/logp, + 2C, log(eC, logp,).
Then the solutions of (4.1) satisfy X< C,.
336 H. M. M. IX WFGER
Ptmji If y 6 $Y, then ~1” > .Y -J> 3~: which contradicts j’ 3 1. So J > 4.~. Put n = log(.u/jl), then
O<A<X/J’- 1 <F ” ,$‘<($r) ” ‘)‘. (4.2)
By s = max( .q y ) 2 p.Fq we obtain
()<A <21- ?)-(I -iiM, (4.3 1
We apply Lemma 2.1 to A, with n = t, q = 2. Since p, 2 3 we have Vi = log pi for i 3 2. Thus
n>exp{-(logX+log(elogp,))C,(l-J)logp,}.
Combining this with (4.3) we find
X~C,log(elogp,)+log2/logp,+CC,logX.
The result now follows from Lemma 2.3, since C, > e2. 1
EXAMPLES. With t=2, 2<p,,<199 and S=$, we have C,<2.3Ox 10” and C,<1.97x 1019. With t=6, 2<p,<l3 and S=+ we find C, < 8.37 x 1O33 and C, < 1.35 x 1036.
4.B. The Case t = 2
In this section we work out the example t = 2, 2 <pi < 199 and 6 = &. We find all solutions of (4.1) with these parameters, thus extending a result of Cijsouw, Korlaar, and Tijdeman (Appendix to Stroeker and Tijdeman [20]). We write
n=Ix,logP,-.Glogp,l,
where x1, x2 are both positive integers. We assume that
p;> 1025, (4.4)
since it is easy to find the remaining solutions. Let logp,/logp, have the simple continued fraction expansion
logp,/logpz = 10; a,, az,...l,
and let the converge& ~,,/q,, be defined by
r -1 -,-- 3 ro - - 0, r”=a,r,-, +r,-,,
4-1 =o, qo= 1, qn=anqn-l+qn-2 (n = 1, 2,...).
DIOPHANTINE EQUATIONS 337
It is well known that r/q is a convergent of a real number CI if
la - rlql < 1/2q2,
and that a11 convergents r,/q,, of a = [ao; a,, a,,...] satisfy
MatI + 1 + 2) qt, < )a - r,/qnl -c l/a,+ lqz. (4.5)
We may assume that (x,, x2) = i. We now have the following criteria.
LEMMA 4.2. Assume (4.4).
(a) If (4.3) holds for some x1,x2, then x2=rk, x,=qk for some k 6 92, and
1 l%P* a,+,+2>py”--. qk 2l”O
(b) If for some k
1 l%P, ak+, >pq’/” - - qk
2’/10 ’
then (4.3) holds for x2 = rk, xl = qk.
Proof: First we show that x, > x2, hence X= x, . Namely, if x, < x2, then
A =xzlogp,-x, logp,>X(logp,-logp,)~Xlog~>O.OIO1 x.
From (4.3) and (4.4) we infer
0.0101 d 0.0101 x< A < 2”‘010-5’2 < 0.0034,
which is contradictory. Next we prove that
p;y”O > 3.1 x. (4.6)
Namely, suppose the contrary. Then 2x”o< 3.1 X, and it follows that X < 80. This contradicts 3.1 X2 pF”O > 105’*. Now, (4.3) is equivalent to
(4.7)
Pl
TABL
E II
(The
orem
4.
3a)
P;’
PZ
.x2
,” cc
P;:
delta
3 ti 31 2 13 2 1 19 2 2 5 13
17
6: : 29
23
11 5 :; :: 11
9 23
23
;: :: 11
12
:: 17
j2
125
128
128
256
343
512
2187
2187
21
97
6859
29
791
3276
8 62
7 48
517
1 12
589
9906
8 42
624
1 62
841
3597
9 10
449
2 21
331
4919
0 66
161
2 25
179
9813
6 85
248
2 25
179
9813
6 85
248
2 38
418
5791
0 15
625
3 93
737
6385
6 99
289
9 90
457
8032
9 05
937
11
3988
9 51
853
7314
3 11
69
414
6092
8 34
141
11
9209
2 89
550
7812
5 11
92
092
8955
0 78
125
12
2005
0 97
657
0582
9 21
91
462
4432
0 20
321
45
9497
2 98
635
7216
1 59
60
464
4775
3 90
625
177
9176
2 17
794
6041
3 35
3 81
478
3205
4 69
041
504
0363
6 19
364
6738
3 50
4 03
636
1936
4 67
383
505
4470
2 84
992
9377
1
3 5 3 11
11 : :z 13
t7’ 83
173
181 i; 149 83
19
83
157 89
19
3 19
7 19
7 :: 199 43
71
:: 1:; 59
16
3
9
: 2 2 2 2 2 4 9 1
1191
3 I
6304
3 2
2522
9 3
2133
1
25
27
121
121
125
243
361
529
2197
2209
22
09
6889
29
929
3276
1 42
241
0276
7 03
549
3904
1 66
161
2 25
229
2232
1 39
041
2 35
124
3277
5 37
493
3 93
658
8805
7 02
081
9 97
473
0326
0 05
057
I1
5149
9 04
768
9841
3 11
51
499
0476
8 98
413
11
6941
4 60
928
3414
1 12
20
050
9765
7 05
829
12
3586
6 42
791
6139
9 21
61
148
2313
2 84
249
0.40
194
0.32
293
P
"0%
~: z r
0.18
716
0.48
703
s 0.
8525
9 m
0.
8089
8 2
0.88
568
0.88
532
% % 0.
7615
9 0.
8794
2 0.
7628
2 0.
8656
0 0.
8759
4 0.
8874
3 0.
8934
3 0.
8986
2 0.
8826
8 0.
8865
6
45
8485
0 07
184
4903
1 0.
8405
9 58
87
158
6708
2 67
913
0.88
64'
174
8874
7 03
655
1304
9 0.
8978
5 35
0 35
640
3707
4 85
209
0.88
568
498
3114
1 43
181
2112
1 0.
8904
0 51
1 11
675
3300
6 41
401
0.89
536
498
3114
1 43
181
2112
1 0.
8958
0
0.000
00
0.21
534
0.48
832
0.28
906
0.40
575
0.22
754
0.46
694
0.49
512
0.45
416
0.29
941
1’: 7 i; 2 2 13
2 ; 2 : 11
3 53 5 1;
2 2 2
41 3
1x : :: 2 2 :
4: : 13
17 :: ;::
60
10
11
17
2:
65
66
42
68 :; 30 f ii :: 15
:z
21
:: :‘4
16
84
84 :i 90
iJ
; 10
2 28
505
4470
2 84
992
9377
1 23
50
5 44
702
8499
2 93
771
59
558
5458
6 40
832
8400
7 41
799
0066
8 57
828
8412
1 79
9 00
668
5782
8 84
121
1152
92
150
4606
8 46
976
1822
83
780
4551
7 61
449
2472
15
921
5084
0 12
303
8650
41
591
9381
3 37
933
9223
37
203
6854
7 75
808
3647
2 99
637
7170
7 86
403
3689
3 48
814
7419
1 03
232
7378
6 91
629
4838
2 06
464
1 09
418
9891
3 15
123
5920
9 2
9514
7 90
517
9352
8 25
856
3 39
456
7389
9 22
223
1484
9 4
9125
8 90
425
6726
1 54
641
9 31
322
5746
1 54
785
1562
5 54
80
386
8577
8 48
021
8593
9 61
32
610
4156
8 09
986
4896
1 94
44
132
9657
3 92
904
2739
2 37
7 78
931
8629
5 71
617
0956
8 37
7 78
931
8629
5 71
617
0956
8
379
2922
7 19
491
5558
8 02
161
2392
99
329
2306
1 75
295
9008
3 24
70
6452
9 07
345
0392
7 04
413
1425
7 60
886
8461
7 89
454
4784
1 21
536
9396
3 07
555
7766
3 10
747
3219
9 05
755
8131
7 97
268
3760
7 98
497
3267
5 80
761
1094
7 11
841
1 21
375
1191
4 21
716
6362
2 74
241
1 93
428
1311
3 83
406
6795
2 98
816
1 93
428
1311
3 83
406
6795
2 98
816
1 93
832
4566
7 68
001
9896
7 96
723
2 25
393
4029
0 69
225
8087
8 63
249
123
7940
0 39
285
3802
7 48
991
2422
4 58
7 44
031
0636
0 42
001
8879
5 53
643
6338
2 53
001
1411
4 70
074
8351
6 02
688
5 07
060
2400
9 12
917
6059
8 68
128
2150
4 15
50
293
2802
6 62
396
2152
6 95
351
0552
1
1::
181
107
199
127 1:;
9:
101 29
191
199 4”:
151 I
181 41
181 17
1:;
163 :: 191
199 3
199
1:: 71
;: 89
1;
11
12 Fl
9 ; 11
2: 10
10
14 ; ;: 2 10
14
10
f; :: 22
14 :: 53
504
0363
6 19
364
6738
3 51
1 11
675
3300
6 41
401
550
3290
3 17
162
4844
1
787
6627
8 37
885
4976
1 80
2 35
917
8476
0 91
681
1151
93
665
7823
5 00
641
1838
45
921
2420
1 54
507
2459
37
419
1553
1 18
401
8594
75
474
8609
3 97
887
9269
03
592
9372
1 91
597
3619
1 31
987
9620
1 91
349
3725
2 90
298
4619
1 40
625
7314
2 41
268
9492
8 26
049
I 10
462
2125
4 11
204
5100
1 2
9755
8 23
267
5799
4 63
481
3 38
298
6815
5 95
733
1731
1 4
8941
5 46
411
9070
5 61
799
9 25
103
1023
1 50
136
2932
1 54
60
999
7061
2 05
831
7732
7 61
62
677
9503
3 67
185
1400
1 93
87
480
3376
4 77
543
0564
9 37
7 38
596
8469
5 57
044
9980
1 37
9 29
227
1949
1 55
588
0216
1
377
3859
6 84
695
5704
4 99
801
2390
72
435
6851
5 13
248
4715
3 24
69
9040
3 56
526
2140
3 03
521
1428
5 52
404
4631
8 60
195
2509
3 21
580
6066
2 62
396
0090
4 07
387
3211
8 38
877
9548
5 51
051
5736
9 98
768
3253
3 36
131
8095
1 12
441
1 23
414
7420
1 97
479
4188
8 22
591
1 93
813
4179
4 57
931
3317
8 02
199
1 93
832
4566
7 68
001
9896
7 96
723
1 93
813
4179
4 57
931
3317
8 02
199
2 25
501
1677
4 16
274
3178
6 82
911
123
6354
1 71
303
1158
3 51
179
8056
1 58
7 32
059
5938
5 49
335
3867
3 30
551
6332
5 11
891
3678
9 38
604
3275
9 54
593
5 07
282
0298
9 53
863
7524
7 83
563
9968
1 L5
49
673
1425
1 78
936
4350
9 93
211
3056
1
0.85
578
0.88
985
0.89
708
0.89
710
0.86
722
0.83
013
0.88
680
0.87
580
0.88
441
0.87
844
0.89
170
0.89
721
0.83
799
0.89
916
0.89
800
0.89
368
g
0.89
828
2 0.
8707
1 2
0.84
941
0.88
788
2 0.
8893
3 m
0.
8939
0 0.
8975
5 0.
8607
8 0.
8931
9 0.
8940
2
0.84
151
0.86
903
0.89
326
0.86
709
0.89
791
0.89
060
0.89
106
: .-
TABL
E III
(T
heor
em
4.3b
)
PI
x 1
P;’
P2
.x2
P;’
delta
: : :: : 8 3 9 10 4 1: 7 1 :“o
11
:: 20
15 ;: 23
54
;: 21
55
2;
32
32
1%
128
128
216
256
343
512
1024
12
96
1728
20
48
2187
21
87
2197
50
625
2 19
936
I 12
589
9906
8 42
624
1 12
589
9906
8 42
624
I 52
168
1143
1 69
024
1 94
619
5068
3 59
315
2 25
179
9813
6 85
248
3 65
615
8440
0 62
976
4 11
724
8169
4 15
651
8 29
350
9467
4 71
872
1oOO
OOOO
oOOo
ooOO
O 11
92
092
8955
0 78
125
18
0143
9 85
094
8198
4 21
91
462
4432
0 20
321
21
9369
5 06
403
7785
6 21
93
695
0640
3 77
856
36
0287
9 70
189
6396
8
9 25
27
15:
121
125
225
243
361
529
tE
1764
20
25
2197
22
09
2209
50
653
2 79
841
I 11
913
0473
1 02
767
0.85
259
1 15
683
1381
4 26
176
0.89
628
1 53
157
8985
2 64
449
0.85
597
1 95
312
5000
0 00
000
0.83
986
2 21
331
4919
0 66
161
0.88
532
3 67
034
4486
9 87
176
0.84
507
4 09
600
0000
0 00
000
0.89
095
8 14
040
6085
1 91
601
0.89
154
9 90
457
8032
9 05
937
0.87
396
12
2005
0 97
657
0582
9 0.
8986
2
17
7147
0 oo
oo0
oooo
o 0.
8909
6 21
61
148
2313
2 84
249
0.88
656
21
9146
2 44
320
2032
1 0.
8169
0 21
61
148
2313
2 84
249
0.88
845
36
5203
4 74
360
5657
6 0.
8873
5
.ooo
oo
0.2
1534
0.
4883
2 0.4
oOOo
0.
2890
6 0.
4057
5 0.
2275
4 0.
4087
6 0.
4669
4 0.
4951
2
0.45
416
0.46
007
0.49
607
0.48
070
0.41
184
0.29
941
0.40
194
0.32
293
0.30
762
0.36
309
19 3 52
35
:: 15
12
11
3: 1:
3:
19 ! 20
14
17
;i 17
:: 49
20
:; 3”:
;:
23
20
: 3’
: 15
25
42
0529
8 34
622
5705
9 50
03
154
5098
9 99
707
51
1858
9 30
140
9075
7 95
42
895
6661
6 82
176
96
5491
5 73
730
4687
5
155
5680
9 55
578
1222
4 50
5 44
702
8499
2 93
771
558
5458
6 40
832
8400
7 78
9 73
022
3053
6 02
816
789
7302
2 30
536
0281
6 79
9 00
668
5782
8 84
121
2481
15
287
3203
7 36
576
6502
11
142
2497
9 47
648
6568
40
835
5712
8 90
625
6568
40
835
5712
8 90
625
3689
3 48
814
7419
1 03
232
2 43
569
2242
1 60
813
0539
7 2
9514
7 90
517
9352
8 25
856
6 72
749
9949
3 25
600
0920
1 9
3132
2 57
461
5478
5 15
625
41
3954
5 12
236
9384
7 65
625
54
8038
6 85
778
4802
1 85
939
61
4094
2 21
446
4815
4 97
216
94
4473
2 96
573
9290
4 27
392
131
0720
0 00
000
0000
0 00
000
188
8946
5 93
147
8580
8 54
784
377
7893
1 86
295
7161
7 09
568
2392
99
329
2306
1 75
295
9008
3 33
25
2567
3 00
796
5087
8 90
625
1978
4 19
655
6603
1 35
891
2397
9 32
199
0575
5 81
317
9726
8 37
607
1 93
428
1311
3 83
406
6795
2 98
816
2 25
393
4029
0 69
225
8087
8 63
249
10
8347
0 59
433
8837
2 20
418
3025
1 22
95
856
9288
6 98
149
5482
2 20
544
171
6155
8 31
334
5863
4 29
238
9520
1 17
1 90
707
9974
8 42
259
1028
6 58
176
171
9070
7 99
748
4225
9 10
286
5817
6 25
251
1682
9 40
423
4886
1 69
433
5937
5
10
42
4207
4 74
827
7657
6
:: ::,
50
5421
0 65
117
2681
7 50
54
210
6513
7 26
817
96
5491
5 73
730
4687
5 97
65
625
WOO
0 00
000
13
154
4723
7 77
391
1946
1 13
50
4 03
636
1936
4 67
383
,iO
3290
3 17
162
4844
1 78
7 66
278
3788
5 49
761
799
0066
8 57
828
8412
1 78
7 66
278
3788
5 49
761
:: 24
72
1592
1 50
840
1230
3 65
82
9520
0 58
400
3528
1 13
65
02
1114
2 24
979
4764
8 12
65
82
9520
0 58
400
3528
1
28
:i
3725
2 90
298
4619
1 40
625
2 44
140
6250
0 00
000
0000
0 2
9755
8 23
267
5799
4 63
481
6 71
088
6400
0 OO
OOO
OOOO
O 9
2510
3 10
231
5013
6 29
321
41
2906
5 87
69.8
35
408
0153
6 54
60
990
7061
2 05
831
7732
7 61
32
610
4156
8 09
986
4896
1 93
87
480
3376
4 77
543
0564
9 13
0 90
925
5398
6 67
734
3846
4
188
3234
9 19
413
1742
6 09
041
379
2922
7 19
491
3558
8 02
161
2390
72
435
6851
5 13
248
4715
3 33
34
4626
7 95
181
5307
0 88
493
1977
9 85
201
4625
5 88
779
3408
1 32
118
3887
7 95
485
5105
1 57
369
1 93
832
4566
7 68
001
9896
7 96
723
:: 2
2550
1 16
774
1627
4 31
786
8291
1 10
84
280
3560
5 96
593
2354
2 07
744
21
22
9468
2 51
895
1294
0 71
398
7276
8
::,
171
7986
9 18
400
0000
0 00
000
0000
0 17
1 61
558
3133
4 58
634
2923
8 95
201
17
171
7986
9 18
400
OOOO
O OO
OOO
000@
3 18
25
259
9333
5 73
498
0608
1 18
208
0664
9
0.87
619
0.88
076
0.88
656
0.88
631
0.88
575
0.87
497
0.85
578
0.89
708
0.85
579
0.89
216
0.89
710
0.86
739
0.89
872
0.89
414
0.85
892
~1
0.89
721
3 0.
8710
1 2
0.89
800
0.87
486
0.89
638
8
0.87
993
3 0.
8873
0 m
g;
;;;
s 5 0:
8686
3 E;
0.88
695
3 0.
8936
8 0.
8707
1 0.
8912
6 0.
8494
3 0.
8939
0 0.
8940
2 0.
8690
3 0.
8799
1 0.
8751
6
0.89
088
0.89
829
0.88
250
0.88
234
&
342 B. M. M. DE WEGER
It follows from (4.6) that
hence .X,/X is a convergent of log p,/log p2, say .x2 = rk, X = qk. Since qk is at least the (k + 1)th Fibonacci number, and by XC 1.97 x lOI (from the examples at the end of Sect. 4.A), we obtain k d 92. The lemma now follows from (4.5) and (4.7). 1
To solve (4.1) we computed the continued fraction expansions and the convergents of logp,/logpz exactly, up to the index n such that q, ~ , < 1.97 x lOI < q,!. Lemma 4.2 guarantees that n 6 93. Doing so, we obtained the result,
THEOREM 4.3. (a) The diophantine inequality
(4.8 1
with p,, p2 primes such that p, <p: < 200, and
x ,, .x,EZ, x,22, .x,>2, andeither S=J
or 6 = 6 and min(p-;I, p-z’) > lOI (4.9)
has only the 77 solutions listed in Table II.
(b) The diophantine inequality (4.8) with p,, pz non-powers such that 2 <p, <p2 < 50 and conditions (4.9), has on1.v the 74 solutions listed in Table III.
In Tables II and III, the column “delta” gives the real number with 1~;’ -p’;21 = min(p.;l, ~;z)~~“~. Note that in Theorem 4.3 we do not demand (xi, x2) = 1. The numerous solutions of (4.8) with 6 = & and min(p.;‘, p?) < lOI can be found without much effort. The computations for the proof of the theorem took 35 sec. We computed approximations of log pi by writing it as a suitable linear combination of numbers of the form log( 1 + x) for small x, and evaluating log( 1 + X) by a Taylor series, taking care to avoid mistakes by rounding-off procedures. Thus we computed explicit rational numbers 8,, 0, with
8, <logp,/logp,<e,<e, +E
for a small enough E. Then as far as the partial quotients of the continued fraction expansions of 8, and l32 coincide, they coincide with the partial quotients aj of log p, /logp,. It appeared to be sufficient to take E = lOPso.
DIOPHANTINE EQUATIONS 343
Note that Lemma 4.2. does not yield a decision if
ak + 1 < pft”’ 1 hP2 --<ak+,+2.
qk 2l”O
Since this gap is relatively small, this situation is unlikely to occur. We met only one such a coincidence, namely for p, = 15, p2 = 23. Here, log15/log23=[0;1,6,2,1,51,... 1, so that a,=51, r,=19, q4=22, and 1522”o & log 19/2 “‘O = 51.4... E [Sl, 53). We have further /1 = 0.002714... < 0002771- = 2’/‘015-22/‘o so (4.3) holds. But (4.1) does not hold, since ldg(1522~2319)/10g(2319j=0.9008... . This example illustrates that (4.3) is weaker than (4.1). Therefore all found solutions of (4.3) have been checked for (4.1) as well.
4.C. The Case t > 3
In this section we show how the L3-BRA can be used to reduce an upper bound for the solutions of (4.1) in the multi-dimensional case. This will enable us to find all solutions of (4.1) for given t >, 3, pl,..., pr and 6.
Let s, v be a solution of (4.1). Put xi = ord,(x/y) (i = l,..., t), and X=max , <, <, IxJ. Let C be an upper bound for X, for example, C = C, (cf. Theorem 4.1). Choose a positive constants y E Z, Co E R, and put
ei = CYCO log Pi1 (i = l,..., t). (4.10)
Consider the lattice Tc Z’, generated by the column vectors of the matrix
Y 0 A= 0 '..
i i
Y 8, ... 8,
Put Ib=x,e, + ... +x,0,. Then
With this notation we have the following useful lemma.
LEMMA 4.4. Suppose that for a solution of (4.1)
14 ’ c lx,1 i=l
(4.11)
344 U.M.M.DEWEGER
COROLLARY 4.5. Let X0 he u positiw number such that
I(f)3(4r’+(t- l);s’)‘?X,,. (4.13)
Then (4.1) has no solutions with for i = l,..., t,
10g(2’-“yC,/t&)/(1-6)10gp,~ Ix,/ <X”. (4.14)
1 4.4. Put A = log(x/y) = XI= 1 x, log pi. Then
I = i .~,(CK, log P,l - YC” l%P;) 6 .i I-y,/9 ,= I ,=I
Proof of Lemma
/A-yC,A
whence, by (4.11),
IAl 2 /;I/ - i l-K,\ ( YC,)>O. r-1
By (4.2) we infer
XC2 JAI- “{’ 6) < (21 fiyco/( 11, -i, ,uK,l))‘i” --n’.
Now (4.12) follows, since pj’gl 6 max(x, y) =x. 1
Proof of Corollary 4.5. By x # y we have y # 0. Suppose that Ix,/ d X0 for all i. Then
l(lJ26 Iy12=y2 ‘x’ ~2 + i,’ < (t - 1) ?‘Xi + A’. i= I
By (4.3) it follows that
A’>l(f)‘-(t- 1)y2i@4t2X&
and we infer
IA/- f, /xi~~2tXo-tX*=tXo. i= 1
Now apply Lemma 4.4, and the result follows at once. 1
DIOPHANTINE EQUATIONS 345
We use the corollary to reduce the upper bound C for X as follows. Choose C,, somewhat larger than (tC)‘. The parameter y is used to keep the “rounding-off error” lyCO log pi - oil relatively small. (If C,, is large, then this error is already so small compared to C, that it is safe to take y = 1.) The ei are integers, and are computed exactly. By the L3-BRA we can com- pute a lower bound for l(r) (cf. Lemma 3.1). We may expect that this bound is of size (det(T)) ‘I’, which is about ytC. Thus we may expect that (4.13) holds with X0= C. Otherwise we may try some larger C,. If (4.13) holds, then (4.14) gives bounds for Jxil, and thus for X, of size log(C,/C), which is of size log C. Hence the reduction of the upper bound is con- siderable indeed. Lemma 4.4 is more precise than its corollary, and therefore more suitable for reducing a small bound C.
We now proceed with an elaborate example. Let t = 6, p, ,..., p6 = 2 ,..., 13, and 6 = I. By the example at the end of Section 4.A, we know that X< C for C = 1.35 x 1036. We take C, = 10Z4’, y = 1. The values of the Bi were computed exactly. We applied the L3-BRA to the corresponding lattice I-,, and found a reduced basis c, ,..., c6 with (c, 1 > 9.40 x 1039. So Lemma 3.1 yields
I( r, ) > 2 ~ 5’2 x 9.40 x 1o39 > 1.66 x 1039.
This is larger than ,,/% C= 1.64... x 1037, so (4.13) holds with X0 = C. Hence, by Corollary 4.5,
X< log(2”2 x 10240/6 x 1.35 x lO36)/$ log 2 < 1350.4,
so X< 1350. Next we choose Co = 1032, y = 1, and C = 1350. The reduced basis of the corresponding lattice T2 was computed, and we found (c, 1 > 2.71 x 105. Hence I( r,) > 4.79 x 104, which is larger than ,,/% C = 1.64... x 104. So (4.13) holds for X0 = C, and Corollary 4.5 yields
lx,1 < 187, 1x21 < 118, 1x31 6 80,
1x41 G 66, 1x51 < 54, IXJ Q 50. (4.15)
Next we choose Co = 1012, y = 104. We use Lemma 4.4 as follows. If IA1 > lo6 then (4.11) holds by (4.15) and (4.12) yields
1x11 d67, ix21 G 42, 1x31 6 29,
ix41 d 24, 1x51 < 19, 1x61 d 18. (4.16)
All vectors in r3 satisfying (4.15) and IL1 < lo6 can be computed with the algorithm of Fincke and Pohst [7] (we omit the details of the com-
346 R. M. M. DE WEGER
putations). We found that there exist only two such vectors, but they do not correspond to solutions of (4.1). Hence all solutions of (4.1) satisfy (4.16). Next we choose C, = lo*, 7 = 104. If /iI > 5 x lo’, then (4.12) yields
TABLE IV (Theorem 4.6)
-1 0
21 1
19 6
-2 11
1 -22
13 1 3
-26 3 8
25 -6
8 1
-4 -4
16 -8 -5
-25 2
- 14 -24
-5
2 18
7 - 10
-11 -1 0 6 0 4 5 I -6 0
-2 -2 -1-3 0 13 -1 -3 -1 -2 0 0 -8 1 0 2 -I 1-6 3
15 -1 -2 -4 0 -15 0 2 I 1
8 -1 -8 0 3 5 1 -I 1 3
1 3 -1 I -6 2 9 -4 -4 0 3 0 4 2 -7 10 530
-13 10 -2 0 0 -2 -10 4 1 1
1 -4 o-5 0 1 -2 -6 0 7
-13 0 3-2 3 -13 -3 7 2 0
-I -4 l-4 I 2 -II 2 6 0
-3 5 1 -1 -6 8 0 -8 3 2
-2 -5 11 0 -3 I 1 o-2 5 0 13 -9 -2 0
19 -2 -4 1 -1 -1 -2 12-l 0
5 10 0 1 -8
-4 -9 3 7 -2 7 0 -13 0 2
-5 3 -9-3 8
17 71561 17 71470 91 17 71875 17 71561 314 20 97152 20 96325 827 31 88646 31 88185 461 51 67168 57 64801 2367 88 58304 88 57805 499
143 48907 143 48180 727 143 50336 143 48907 1429 288 29034 288 24005 5029 293 62905 293 60128 2777
337 92000 337 87663 4337 351 56250 351 53041 3209 627 52536 627 48517 4019 671 10351 671 08864 1487 781 25000 781 21827 3173 878 95808 878 90625 5183
1006 63296 1006 56875 6421 1882 45551 1882 38400 7151 1929 14176 1929 13083 1093 1992 97406 1992 90375 7031
4392 39619 4392 3OOKI 9619 7812 58401 7812 5oooO 8401
14336 00000 14335 62273 37727 14758 24779 14757 89056 35723 19773 26743 19773 oooo0 26743 40600 88955 40600 86272 2683 48828 12500 48827 86447 26053
1 27848 76137 1 27848 44800 31337 1 38412 87201 1 38412 03200 84001 2 61035 15625 2 61033 83072 I 32553
2 67363 98612 9 68892 08832
1305 16915 36COO
2 67363 28125 70487 9 68890 10407 1 98425
1305 16881 72831 33 63169 2834 49760 00000 41 04623 10 -6 5 -6 4 2834 49801 04623
DIOPHANTINE EQUATIONS 347
There are 143 vectors in r4 satisfying (4.16) and 121 < 5 x 105. Of them, 2 correspond to solutions of (4.1), namely the vectors with
(x ,,..., x6) = (7, -5, 3, -9, -3, 8) ,I = 257674,
(x , ,..., x6) = (- 10, 10, -6, 5, -6, 4) A= 144817.
Both also satisfy (4.17). Hence all solutions of (4.1) satisfy (4.17). At this point it seems inefficient to choose appropriate parameters Co, y
to repeat the procedure with. But the bounds of (4.17) are small enough to admit enumeration. Doing so, we found 605 solutions of (4.1). We cannot list them all here. Instead we give the following result, from which the reader should be able to find all solutions without much effort.
THEOREM 4.6. The diophantine inequality
o<x-y<y’J’
in x, yES={2.“... 13”?xi~Z2, xi20 (i= l,..., 6)) with (x,y)= 1 has exactly 605 solutions. Among those, 571 satisfy
ord,(xy) < 19, ord,(xy) d 12, ord,(xy) < 8,
ord,(xy) d 7, ord,,(xy) < 5, ord,,(xy) d 5.
The remaining 34 solutions are listed in Table IV.
The computation of the reduced basis of rI took 113 set, where we applied the L3-BRA as we described it in Section 3, in 12 steps. The direct search for the solutions of (4.17) took 228 sec. The remaining computations (computation of the log pi up to 250 decimal digits, of the reduced basis of r2, and of the short vectors in r3 and r,) took 8 sec. Hence in total we used 349 sec. The numerical details can be obtained from the author.
5. A DIOPHANTINE EQUATION
Let p1 < ... <pr be prime numbers, where t > 3, and let S be the set of all positive rational integers composed of those primes only. In this section we study the exponential diophantine equation
x+y=z (5.1)
in x, y, z E S. Without loss of generality we may assume that x, y, z are relatively prime. For any a E S we define
m(u) = ,yy, or-d,(a). .,
64112613-S
348 R.M. M. DE WEGER
It was proved by Mahler that m(xyz) is bounded for the solutions of (5.1). An explicit bound can be computed (cf. Shorey and Tijdeman [ 18, Corollary 1.21). We do so in Section 5.A. In Section 5.B we introduce multi-dimensional p-adic approximation lattices, and in Sections 5.C and 5.D we show how to reduce the found upper bound, and to solve (5.1) completely, in the cases t = 3 and t 3 4, respectively. We conclude with some remarks on a conjecture of Oesterlt and Masser, which is related to Eq. (5.1) in Section 5.E.
5.A. Upper Bounds
THEOREM 5.1. In the above notation, put
s = [2t/3], p=p, “‘P,,
V, = max(e, log pi) (i’ I,..., t), Q= V,p,r+,.‘. v,,
G,=2, G,=6, GDx=p,- 1 ifp,>5, G= ,~~~tGp,llOg~i, .-.
C, = 29” + 26ss+ 4Q log(e V, , ).
Choose p, K with 2/(s + 1) < p 6 2, 0 < K < p/2, and put
E=(P--K)/(l +K)(l +PL)(J+ I),
k=max{(l6s) (I+l/X)(S+-I), (8/E)(1+Pc)(S+i), 161/C},
C,=4(s+ I)(s+‘)kcl+p’GfZ, ~,=4(~,+~~~~~(~/P,))fl~~Pl~
c, = Cdl% c, J2, C,. = max( C,, C,(log C,)2).
Then all solutions of (5.1) satisfy m(xyz) < Cl,,.
ProoJ If we consider instead of (5.1) the equivalent equation
x f y = 2, (5.2)
then we may assume that xy has at most s prime divisors. Suppose that m(xy) < 7. Then
hence
py(‘) d z < 2 max(x, y) < 2(P/p, I’,
m(z) G ww/P,)‘)/l% PI 3
and it follows that m(xyz) < C,,. Now let m(xy) 2 8, and suppose m(z) > 2. Then for some p =pi,
m(z) = ord,(z) = ord,( +x/y - 1) = ord,(log,(x/y)).
DIOPHANTINE EQUATIONS 349
Put x/y = pi, +*..,>. Then m(xy) =max((xi,l,..., IxJ). On applying Lemma 2.2 we obtain
m(z) < C,(log m(xy))2. (5.3)
Obviously (5.3) is also true if m(z) < 2. If in (5.2) the plus sign holds, then
(P/p,)““‘> z > max(x, y) >pyC”?‘).
By (5.3) it then follows that
MXY 1-c G l”z;E1) (log m(xy))! (5.4)
If in (5.2) the minus sign holds, then we apply Lemma 2.1 as follows. Suppose that
4XY) l%P, 3 (Cc + c7 logwP,)wg WY))‘. (5.5)
Then it follows that
C,~log~(xY))210g(~/P,)6~(xY)logP,-log2.
Note that by (5.3)
so that /y/x- II<+. Hence
I 1 WA) C7hw(xv)Y
llog(Ylx)l G 2 log 2 5-l 62log2 mkv) Pl
On the other hand, Lemma 2.1 yields
lb3(ylx)l >expi - G(log I + log(e~,))}.
Thus we obtain
WY) l%P, < wag wY))2 1%wP,) +log(2 log 2)
+ G(log m(v) + log(eV,)).
Obviously,
G(log m(w))* > log(2 log 2) + G(log WY) + log(eV,)),
350 B.M. M. DE WISER
and we have a contradiction with (5.5). So from (5.4) or from the negation of (5.5 ) we infer
and from Lemma 2.3 we obtain rn(-~y) < C,. Now the result follows from (5.3). I
EXAMPLES. With t = 3, p, = 2, pz = 3, p3 = 5 we find a minimal value for k i+lc on taking p= 1, K-A, namely k’+jf=2’08. Then C,,<6.75 x 104’. With t=6, p ,,..., pb=2 ,..., 13 we take p=l, ti=+, and we find c <337x 107’. 10 .
5.B. Approximation Lattices
In [23] the concept of (2-dimensional) approximation lattices of a p-adic number was introduced. In this subsection we generalize this notion to multi-dimensional approximation lattices of a linear form of p-adic num- bers. We confine ourselves to the particular lattices that we use for solving Eq. (5.2), and indicate how a basis of such a lattice can be computed explicitly.
Let p be any of the primes p, ,..., pt. We may assume that p [ ~JY Rename the other primes as po,...,p,~2, such that ord,(log,(p,)) is mmimal. For i = l,..., t-2 and mEN, put
% m-l
@;= -lo~p(P~)llo~p(PO) = 1 ui.l P’, l9j’“’ = c u,.,p’, /= I I= I
where u,,, E (0, l,..., p - 13. Then 0, is a p-adic integer by the choice of po, and 6!“’ is the unique rational integer satisfying ord,((!Ij - Sj”)) > m and 0 d 0:‘) <pm. The Ojmr can be computed for the desired m by using the Taylor series for the p-adic logarithm,
log,(~)=$log,(I+(%~-l))=~ i (-1)“’ (Xk - 1 ,‘/I, /= I
where k is the smallest positive integer such that ord,(Xk - 1) 2 1. Consider the lattice r, c Zr-’ generated by the column vectors
b ,,..., b,_*, b, of the matrix
DIOPHANTINE EQUATIONS 351
Put m, = ord,(log,(p,)). Then
rt?l= {(x I)...) X,&Z, x0) E Z’- l: 1x,8,+ .‘. +X,_*e,-*-X&<p-m}
= {c%...,xt-2r x,)EZ’-‘: (log,(p~...p:L-:)l,~~--‘“+“o’}.
We call such a lattice an approximation lattice of the p-adic linear form x, 8, + . . . + X, _ 2 6, ~ 2. For t = 3 we have exactly the approximation lattice of 0, in the sense of [23]. (Note that there is a different matrix notation there). Further, consider also the set
It is clear that rz is a sublattice of r,,,. In general, the two are not equal, since (x ,,..., x,+~, x0) E r,,, only implies p$. . . p::; = i (mod pm +“‘O) for some (p - 1)th root of unity 5, not necessarily + 1. (Recall that log,([) = 0 if and only if [ is a root of unity). For p = 2, 3 the only roots of unity in Q, are +I, so then rz=r,,,.
For p > 3 we show how a basis b:,..., b:- *, b,* of rz can be computed from a known basis b, ,..., b, ~ 2, b, of r,,,. Let < be a primitive (p - 1)th root of unity in Qep. For any x = (x, ,..., x, ~ 2, x0) E r, we define k(x) E: Z by
P?.. p:L-; E jW (mod pm + "o), O<k(x)<p-2.
Then k( x ) is (mod( p - 1)) a linear function on r,, and x E f 2 if and only if $(p - 1) ( k(x). Put
k = gcW(bo),..., k(b, ~ 2)h
and compute (by the euclidean algorithm) a basis b;,..., b;- 2r bb of r, such that k(bb) = k. Put for i = l,..., t - 2,
y,r k(bj)/k (mod(p - 1)/2), IYil 6 (P- I)/43
b* = b/ - y,bb.
Then k(bT) z k(bj) - yik(bb) = 0 (mod(p - 1)/2) (i = l,..., t - 2). Put
y. = lcm(k (P - I)/2 )lk,
which is the smallest positive integer such that y,k = 0 (mod(p - 1)/2). Every x E r, can be written as
x=y,b:+ ... +y,~zbT_2+yob;, Y,E&
since bf ,..., b:_ 2, bb is a basis of I-,,,. Now,
k(x) = y,k (mod(p - 1)/2).
352 B. M. M. DE WEGER
So x E fz if and only if ‘J(, ( yo, Hence put
then it follows that b:,..., bT_ 2, b,* is a basis of r,$ In practice it may occur that p0 can be chosen such that it is a primitive root (modp). Then choose i 5 p0 (mod p), and it follows from k(b,) = 1 that b: = b, (i = O,..., t - 2). If pi sp; (mod p), then
m-1 y,ra,+8]“‘=ai+ 1 u,,, (mod(p- 1)/2) (i= l,..., t-2).
/=O
Yo = (P - 1 J/2.
So in this special case it is simple to find a basis of L’,$ For any solution x, y, z of (5.2) put x/y =p$ . . p;‘~:;, and consider the
point x = (x, ,..., xrM2, xO)e Z’- ‘. Suppose that ord,(z)82. Then
ord,(z) = ord,(x/y k 1) = ord,(log,(x/~v))
=ord,(x,8,+ ... +~,~~~,~~-x~)+m~.
Hence x E rz for m < ord,(z) - m,. With this notation we have the follow- ing useful lemma:
LEMMA 5.2. Let me N and X,>O be constants such that
l(l-2)> (t- l)“Z x0.
Then (5.2) has no solutions x, y, z with
(5.6)
m + m, 6 ord,( z) 6 m(xyz) < X0. (5.7)
ProoJ Suppose (5.7) holds. By ord,(z) B m + m, we have x E I% x # 0. Further, ixil < m(xyz) < X0 (i = O,..., t - 2). Hence
r-2
l(r;)2dIx12= c xfd(t-1)X& r=O
which contradicts (5.6). 1
Suppose that we know that m(xyz) < X0. We may expect that l(rz) is of size (det(rz))“‘‘-*‘, which is about p *A- ’ ), Thus we may expect that it
will suffice to take m somewhat larger than (t - 1) log(m X,)/log p. If (5.6) does not hold then, we may try some larger m. If (5.6) holds, then (5.7) yields ord,(z) <m + mo. We repeat this procedure for p =p, ,..., p,. Since (5.2) is invariant under permutation of x, y, z we find a new upper bound for m(xyz), which is of size m N log X0.
DIOPHANTINE EQUATIONS 353
5.C. The Case t = 3
We illustrate the use of the p-adic analogue of the one-dimensional con- tinued fraction algorithm by solving the equation
xfy=wz, (5.8)
wherex,y,zE(pk:p=2,3,5,kEiZ,k~0},andwEZ, lw1<106,(w,z)=1. Put X = maxp = 2,3,5 ord,(xyz). The example at the end of Section 5.A shows that in the case 1 WI = 1 we have. X-C 6.75 x 104’. It can be checked without difficulties that the effect of the w with IwJ d lo6 can be neglected, so that for the solutions of (5.8) also X < 6.75 x 104’ holds. Put for p = 2, 3, or 5,
such that
0 = -log, PI /log, PO
is a p-adic integer. Then define the lattices r,,, and rz as in Section 5.B, so r,,, is generated by
b,=(D:“,). bo=(,“:)
For p = 2, 3, we have r; = r,, and for p = 5 a basis of r;? is
b:=b,-yb,,, b,* =2b,,
where y = 0 if t? (m) is odd, y = 1 if 8 ‘m) is even. Using the algorithm of [23, Sect. 31, we can compute a basis c, , c2 of rz that is reduced in the sense that
Cl.1 c, = ( ) Cl.2
has minimal norm @(c,)=max(lc,,,l, Jc~.~/) in r:\(O). We choose p, po, p,, and m as in the foliowing table, where m is chosen so that pm is somewhat larger than (6.75 x 1041)‘:
P p. pi m. m y O(q)> u< w 1x01 d 1x11 G
2 3 5 2 297 2 148 298 1O6x2298 222 152 3 2 5 1 189 3 94 189 lo6 x 3’89 354 152 5 2 3 1 135 0 567 135 lo6 x 513s 370 233
We give the values of eCrn) in Table V, and the reduced bases of the rz in Table VI. From this table we find the lower bounds for @(c,) given above.
TABL
E V
(Sec
tion
5C)
(p-a
dic
nota
tion:
O
.abc
...
= a
+ b.p
+
c.p2
+ .
)
thet
a =
-log,
5/
lag,
3
= 0.
1010
1 11
101
cm01
11
110
llooo
10
101
OOOC
Q 01
001
1110
1 00
010
1oOG
.I 10
011
1011
0 10
000
0101
1 11
100
OOOo
l 11
010
@Jo
00
0000
1 00
010
1110
0 11
100
1011
1 01
001
0110
1 11
m
0101
0 01
110
0101
0 11
110
OwoO
00
101
0011
0 00
100
0001
1 01
111
0111
0 01
010
1110
1 10
010
0100
1 00
001
1010
0 00
111
0000
1 11
111
0111
1 00
011
1011
0 OO
OOO
0010
1 01
101
0110
0 00
010
0111
0 11
100
1110
1 11
011
0101
1 10
1 11
o.
...
thet
a =
-log,
S/
log,
2
= 0.
1102
2 12
121
2200
1 12
010
2110
2 10
210
1002
2 20
210
2001
0 10
112
2220
1 21
021
2102
2 10
000
2202
0 12
012
0202
2 21
001
0001
2 02
020
2121
0 12
202
1220
0 00
000
1012
0 00
211
1202
1 10
120
0210
0 10
222
2212
2 01
201
2111
1 11
121
1100
1 20
222
1000
0 20
121
2?22
1....
thet
a =
-log,
3/
lag,
2
=
0.33
002
0200
3 04
411
2312
0 44
012
0101
1 00
044
4320
4 30
340
0002
3 14
333
1241
3 43
420
4030
2 10
202
4410
4 32
433
-144
32
030-
71
1231
1 34
044
4023
1 04
112
3323
0 00
242
1423
2 14
400
3110
....
TABL
E VI
(S
ectio
n 5C
)
p =
2(ba
se-2
no
tatio
n)
(
-101
11
001
0001
1 11
100
1010
1 01
010
0111
0 00
101
1111
0 10
000
1100
0 11
100
1001
1 11
001
0111
0 10
000
1000
1 00
011
1111
0 01
100
1001
1 01
101
0100
0 11
110
0100
1
b,=
1011
1 10
010
1100
1 10
111
-101
0 11
101
0101
1 00
011
0000
0 00
111
0100
1 11
001
1001
0 10
110
0011
1 00
001
0110
0 11
101
0001
1 01
001
0010
0 10
101
1101
1 10
011
ml1
01
011
0101
1 00
110
1111
1 11
110
1010
0 01
010
0110
1 00
011
1011
lo
o01
1010
1 01
101
1011
1 10
101
0111
1 10
111
0000
1 01
010
1101
0 10
001
1001
1 11
111
0110
1 10
110
0110
0 11
001
0010
1 lo
000
0001
1 00
100
0101
0 00
101
0101
1 10
000
0011
1 01
100
1010
1 10
110
bz =
-1
010
0110
1 01
110
1111
0 10
000
OOOO
O 10
001
llooo
10
111
0010
0 01
010
1100
0 11
110
1110
1 10
101
1111
0 00
011
1010
1 01
000
1101
0 01
001
0110
0 10
111
0011
0 10
101
0111
0 11
100
0001
0 11
110
1001
0
p =
3 (b
ase-
3 no
tatio
n)
3
b,
= 11
10
0010
2 10
211
2201
0 20
100
1002
2 10
001
2022
0 12
202
0110
2 11
202
1111
0 20
112
2210
1 20
022
2211
2 10
020
1002
2 12
000
%
2000
11
110
2111
1 10
112
1010
2 10
210
2210
1 01
021
1021
2 10
020
1100
2 12
222
2001
2 11
101
0122
2 11
211
0120
1 22
201
02oo
o
bz=
(-120
01 21
02
2111
2 02
001
2210
1 01
111
0022
2 12
000
2002
2 02
011
1220
0 21
221
0020
0 00
120
2021
0 s
0110
0 12
216
1022
2 00
022
2020
2 =!
10
002
2101
0 00
220
1020
0 02
122
0022
1 22
120
2210
0 10
120
1202
0 20
120
2201
2 10
011
1002
2 01
122
2222
2 22
212
0200
1 iF
.4
p =
5 (b
ase-
5 no
tatio
n)
m
b, =
-2
13
2104
1 20
044
2101
1 03
000
0042
0 40
302
1314
4 33
303
4214
3 22
021
3123
3 42
233
4231
4 -1
40
0122
1 40
144
1032
3 41
221
1011
3 13
410
4414
4 41
032
2113
1 43
034
4032
2 11
323
4302
2
b2 =
-2
00
2001
2 43
403
1323
2 12
424
4410
2 00
032
4232
1 20
012
1413
4 22
130
2010
3 00
020
1330
1 23
3 01
424
4201
3 24
004
4324
4 32
120
3023
0 23
141
2234
0 40
304
3111
3 30
442
3344
3 20
012
356 B. M. M. DE WEGER
They are all larger than 6.75 x 10“‘. Hence for the solutions of (5.8) we have u <m + m,, and Iu’I z 6 W, as shown in the table above. We now find the new upper bounds for IxOI, lx,/ as follows. If in (5.8) the minus sign holds, then, on supposing that min(x, y) > W”19, we infer
Ix-),1 = IkV 16 W<rnin(,~,4:)~“~.
By Theorem 4.3a and Table II, the inequality Ix - yl < min(x, Y)~“’ has no solutions with min(x, y) > W, since W> 10’“. Hence min(x, v) < W10i9, and we infer
max(x, y) < min(x, y) + 1~11 z < W’O” + W.
If in (5.8) the plus sign holds, this inequality follows at once. So now the bounds for 1x01, lx11 follow from
lxij log pi < log max(x, y) 6 log( W1o’9 + W).
We repeat the procedure with m as in the following table:
P m Y @(Cl)> UG W 1x01 6 I-~‘1 6
2 17 260 18 106x2’* 31 21 3 13 531 13 106x 3” 49 21 5 8 1 818 8 lo6 x 58 49 31
The numbers are now so small that the computations can be performed by hand. For example, for p = 5 the lattice f,* is generated by
b: = (-3:,,,7)9 b,* = i7xp250)~
and a reduced basis is
c’=( ,:k), c2=(y$ Now, in all three cases, W”” < 1015. On supposing min(x, y) > 1015 we infer
lx-y1 = Iwl 26 W< 1015X9~10<min(x,y)9~10.
By Theorem 4.3a and Table II we see that the inequality Ix - yl < min(x, Y)~“’ has only two solutions: (x, y) = (265, 5**), (284, 353). However, both have Ix -yl > 10” x9’*o. So we infer min(x, y) < 1015, hence by max(x, y) d 1015 + W we obtain the bounds for 1x01, (~‘1 as given above.
DIOPHANTINE EQUATIONS
TABLE VII (Theorem 5.3)
p=2,p,=3,p,=5
357
XI -
P;’
2 9 10 9165625 10 59049 10 9765625 4 81 12 244140625 6 729 10 9765625 2 9 8 390625 6 729 8 390625
10 59049 8 390625 14 4182969 10 9165625 4 81 8 390625 0 1 8 390625
8 6561 8 390625 0 1 6 15625 4 81 6 15625 8 6561 6 15625 6 129 6 15625 2 9 4 625 2 9 6 15625 0 1 4 625 4 81 4 625 0 1 2 25
2 9 1 3 1 3 2 9 3 27 4 81 4 81 6 729 6 729 3 27
5 243 5 243 I 2187 6 129 7 2187
11 177147 3 27 8 6561 I 2187 8 6561
2 1 3 0 1 0 2 2 4 3
3 1 5 0 1 1 5 0 3 4
25 5
125 1 5
25 25
625 125
125 5
3125 1 5 5
3125 1
125 625
sign u w
-1 4 -610351 -1 4 -606661 -1 9 -416831
-1 5 -305153 -1 3 -48827 -1 3 -48737 -1 3 -41447 -1 7 - 38927 -1 4 - 24409 -1 5 - 12207
-1 6 -6001 -1 3 - 1953 -1 3 - 1943 -1 3 -1133 -1 4 -931 -1 3 -17 -1 8 -61 -1 4 -39 -1 5 -17 -1 3 -3
-1 4 -1 1 3 1 1 I 1
-1 3 1 1 5 1
-1 4 5 -1 3 1 -1 6 11 -1 3 13
1 3 19
1 4 23 1 3 31 1 6 83
-1 3 91 1 4 137 1 10 173 1 4 197
-1 5 205 1 3 289
-1 4 371
Table continued
358 B. M. M. DE WEGER
TABLE VII (Theorem S.3)pCon~inud
I 3 5 243 9 19683 8 6561
10 59049 5 243 9 19683 9 19683
IO 59049 I2 531441
I 3 9 19683
13 1594323 IO 59049 10 59049 12 531441
3 27 I 2187
11 177147 3 27
II 177147 11 177147 12 531441 12 531441 12 531441 II 177147 13 1594323 1 2187
14 4782969 13 1594323
13 1594323 1 3 5 243 9 19683
14 4782969 13 1594323 15 14348907 14 4782969 14 4782969 14 4782969
16 43046721 9 19683
15 14348907
3125 3125
125 25
15625 78125
-1 -1
1
5 2 4
4 0
9
9 2
8 4 0
0 11
3125 25
625
78125 78125 78125
625
390625 78125 78125
3125 1953125
125 78125
-I
-I -I
-I 15625
25 1953125
125 1953125
25
3125 1953125 1953125 1953125
I5625 1953125
3125 390625
625
-1
-I
48828125
3 391 3 421 5 619 3 817 5 I357 5 2449 3 2461 3 2851 4 3689 7 4147
4 4883 4 6113 8 6533 3 7303 3 7381 4 8801 3 9769 3 10039 4 11267 7 15259
3 22159 3 31909 4 33215 3 64477 3 66427 5 66571 4 99653 4 122207 5 149467 3 199291
3 199681 3 244141 3 244171 3 24660 1 4 297959 3 443431 5 448501 3 549043 3 597793 3 597871
6 672605 6 763247 4 896807
Table continued
TABLE VII (Theorem 5.3+Continued
p=3,po=2,p,=5
14 16384 10 9 512 9 4 16 8
12 4096 6 7 128 5 2 4 4 1 2 2 5 32 1 6 64 3
11 2048 4
9765625 1953125 390625
15625 3125
625 25
c
12; 625
9 512 10 1024
3 8 15 32768 14 16384 17 131072 16 65536
8 256 19 524288 18 262144
1 25
15625 125
5 78125
3125 78125
25 1
23 8388608 1 5 13 8192 8 390625 22 4194304 8 390625 10 1024 11 48828125 18 262144 9 1953125 20 1048576 4 625
0 1 9 1953125 21 2097152 6 15625
5 32 10 9765625 24 16717216 3 125
23 8388608 10 9765625 26 67108864 7 78125
-1 4 - 120361 -1 3 -72319 -1 3 - 14467 -1 3 - 427 -1 4 -37 -1 3 -23
1 3 1 -1 3 1
1 3 7 1 5 11
1 3 19 -1 3 37
1 4 193 -1 4 403
1 3 607 -1 3 1961
1 3 2543 1 3 2903 1 4 6473
-1 3 9709
-1 6 11507 1 3 14771
-1 5 15653 1 7 22327 1 4 27349
-1 3 38813 1 3 72338 1 3 78251 1 3 361691 1 3 621383
1 3 672379 1 4 829469
p=5,po=2,p,=3
12 4096 16 43046721 -1 3 -344341 5 32 15 14348907 -1 3 -114791 7 128 1 3 -1 3 1 6 64 8 6561 1 3 53
14 16384 2 9 -1 3 131 13 8192 9 19683 1 3 223 20 1048576 10 59049 1 3 8861 21 2097 152 3 27 -1 3 16777
360 B. M.M. DE WEGER
Those bounds are small enough to admit enumeration of the remaining cases. Thus we obtain the following result.
THEOREM 5.3. The diophantine equation
where x = p$, y = p ;I, = = p”, (P, po, P[) = (2, 3, 5), (3,2, 5), or (5,2,3), -x0, x, , u are nonnegative integers, M’ E L, 1 w( < 106, and p l w has exactly 29 1 solutions for p = 2, 412 solutions .for p = 3, and 570 solutions for p = 5. In Table VII all solutions with u > 3 are given.
The computer calculations for the proof of this theorem took 3 sec.
5.D. The Case t 3 4
In this section we present an elaborate example of the use of the L3-BRA for solving an equation of type (5.2) in the multi-dimensional case. Let S be the set of positive integers composed of the primes 2, 3, 5, 7, 11, 13 only. In the example at the end of Section 5.A. we have seen that the solutions x, y, ZE S of (5.2) satisfy m(xyz)< 3.37 x 1073. We show how to reduce this bound, and thus we are able to find all solutions. With the notation of Section 5.B we choose the following -parameters:
P PO PI Pz P3 P.% m. m 70 YI Y1 Y3 Y4
2 3 5 7 11 13 2 1320 3 2 5 7 11 13 1 840 5 2 3 7 11 13 1 600 2 1 0 0 1 7 3 2 5 11 13 1 480 3 0 O-l 1
11 2 3 5 7 13 1 360 5 -2 -1 2 0 13 2 3 5 7 11 1 360 6 3 1 -2 1
We computed the six values of the 0j”’ (i= 1, 2, 3,4), and the reduced bases of the six lattices r,$,. Thus we obtained
P l(r, *)> lc,l/4> ord,(xyz)d
2 6.34 x 107’ 1321 3 2.50 x 1O79 840 5 2.02 x 1os3 600 7 2.39 x 1oso 480
11 2.28 x 1O74 360 13 4.23 x 107’ 360
DIOPHANTINE EQUATIONS 361
These lower bounds for l(r2) are all larger than ,,,$ x 3.37 x 10”. So we may apply Lemma 5.2 with X0 = 3.37 x 1073, which is the theoretical upper bound for m(xyz). For every p we thus find ord,(z) <m+m,. Since Eq. (5.2) is invariant under permutations of x, y, z, we even have ord,(xyz) < m + m,, as shown in the above table. Hence m(xyz) < 1321.
We repeated the procedure with X0 = 1321 and m as in the following table. After computing the reduced bases of the six lattices rz we found the following data (Note that in all cases l(Z-2) >fix 1321.)
P m YO YI Y2 Y3 y4 /(Cl) > ord,(xyz) <
2 77 8342 78 3 49 9026 49 5 35 2 0 1 1 0 22325 35 7 28 3 0 -1 1 0 14403 28
11 21 5 1 1 1 -2 5162 21 13 21 6 0 0 1 2 14779 21
Hence m(xyz) < 78. Next, we repeated the procedure with X0 = 78, and m as in the following table. We found
P m y. yI y2 Y3 Y4 Z(rz) > ord,(xyz) 6
2 55 364 56 3 35 301 35 5 25 2 1 1 1 0 622 25 7 20 3 1 -1 1 0 693 20
11 15 5 1 2 -2 -2 192 15 13 15 6 1 0 3 2 658 15
Hence m(xyz) < 56. To find the solutions of (5.2) with ord,(xyz) below the bounds given in
the above table, we followed the following procedure. Suppose that we are at a certain moment interested in finding the solutions with ord,(xyz) <f(p), where f(p) is given for p = 2,..., 13. Choose a p and an m<f(p) - mo, and consider the lattice l$. If a solution x, y, z of (5.2) exists with ord,(z) 2 m + m,, then the vector
Xl
xc : (9 x4
x0
362 B. M. M. DE WEGER
TABLE VIII (Section 5D)
P m n P m n
2 3 5 I
II 13
2 3 5 I
11 13 2 3 5 7
11 13
2 2
2 2 2 2 2 2 2 2 2 3
3 3 3 3 3 3 5 5
44 28 20 16 12 12 33 21 15 12
- 7 7
11 11 11
9 9
22 14 10
8 6 6
21 20
13 13 13
- 2 2 2 2 2 2 2
19 18 17 16 15 14 13 12 11 13
- 3 1 3
2 5 2 5
5
12 .- 5 11 7
10 1 7
9 1 7 8 1 11 I 6 11
9 - 13 8 13
7 6 5 7 6 5 4 5 4 3
5 4 3
10 9 8 7 6 5 4
3 6 5 4 3 2 1 4 3 2
1 3 2 I 2 I 2 1
6
1 4 -.
1 4
1 2 3 6
15 16 26 31
44 5 8
16 35 54 87
1 5
18
36 -
6 18
1 8 -
4
(n = number of solutions found.)
DIOPHANTINE EQUATIONS 363
with x, = ord,,(x/y) (i = O,..., 4) is in the lattice. Its length is bounded by (f(Po)*+ ... +f(p4) 1 . * “* All vectors in rz with length below this bound can be computed by the algorithm of Fincke and Pohst [7] (we omit details). Then all solutions of (5.2) corresponding to lattice points can be selected. Then we replace f(p) by m + m, - 1, and we may repeat the procedure for newly chosen p, m.
We performed this procedure, starting with the bounds for ord,(xyz) given in the above table for f(p), and with p, m as in Table VIII. At the end we have f(2) = 4, f(p) = 1 for p = 3,..., 13. The remaining solutions can be found by hand. Thus we obtained the following result.
THEOREM 5.4. The diophantine equation
x+y=z
in x, y, ZES= {2”‘... 13”6: x~EZ, xi>0 (i= l,..., 6)) with (x,y)= 1 and x 6 y, has exact1.v 545 solutions. Of them, 514 satisfy
ord,(xyz) 6 12, ord,(xyz) 6 7, ord,(xyz) 6 5,
ord,(xyz) < 4, ord,,(xyz) < 3, ord,,(xyz) < 3
The remaining 31 solutions are given in Table IX.
The computer calculations for the proof of this theorem took 2856 set, of which 2830 set were used for the first reduction step. In this first step, we applied the L3-BRA in 12 steps (cf. Sect. 3) which costed on average about 400 sec. The remaining 430 set were mainly used for the computation of the 24 8j”)‘s. Full numerical details can be obtained from the author.
5.E. Examples Related to the Oesterk-Masser Conjecture
Let x, y, z be positive integers. Put
G= n p. p I x.vz
p prime
For all x, y, z with (x, y) = 1 and
x+y=z
we define
c(x, y, z) = log z/log G.
364 B. M. M. DE WEGER
00C40N000c-l- 0000 -00000 rrmooorl-oom -
3000000~~-0 o--oomo--0 OOOdONOONO 0
b--0-0000- NC-IO- OONPlOO 0000 -oocfoo -
0m0m000000 o-00m000miD 0-60000-00 0
000N006000 oowoo-o+oo -000000000 0
30000606’~ 000m000mb0 00*0000000 0
DIOPHANTINE EQUATIONS 365
TABLE X (Section 5E)
x Y z 44 Y, z)
121=112 48234315 = 3256?3 48234496 = 2=‘23 1.62599 1 4314 = 2 37 4375 = 547 1.56789
343 = 13 59049 = 3 ‘O 59392 = 2”29 1.54708 198425 = 5=7931 96889010407 = 713 96889208832 = 2’*3’13’ 1.49762
121=11’ 255879 = 3913 256000 = 2”53 1.48887 37 32768 = 215 32805 = 3*5 1.48291
3200 =2’S 4823609 = 7641 4826809 = 136 1.46192 1 2400=2s3 52 2401 = I4 1.45567
702021632 = 21913 103 1977326743 = 7” 2679348375 = 3”531 1” 1.45261 1 512000 = 2125’ 512001 = 357243 1.4433 1 1 19140624 = 2437547 19140625 = 5V 1.43906
7168 = 2r07 78125 = 5’ 85293 = 3*13 1.43501 3 125=5’ 128=27 1.42657 5 177147=3” 177152 =2”‘173 1.41268
Recently, Oesterle posed the problem to decide whether there exists an absolute constant C such that c(x, y, z) < C for all x, y, z. Masser conjec- tured the stronger assertion that c(x, y, z) < 1 + E, when z exceeds some bound depending on E only. For a survey of related results and conjectures see Stewart and Tijdeman [19].
It might be interesting to have some empirical results on c(x, y, z), and to search for x, y, z for which it is large. From the preceding sections it may be clear that such x, y, z correspond to relatively short vectors in appropriate approximation lattices. As a byproduct of the proofs of Theorems 4.6 and 5.4 we computed the value of c(x, y, z), corresponding to many short vectors that we came across in performing the algorithm of Fincke and Pohst. All examples that we found with c(x, y, z) > 1.4 are listed in Table X. Our search was rather unsystematic, so we do not guarantee that this list is complete in any sense. The largest value for c(x, y, z) that occurred is 1.626, which was reached by
x= 112, y=32x56x73, ~=2~‘x23.
These results do not seem to yield any heuristical evidence for the truth or falsity of the above mentioned conjecture.
ACKNOWLEDGMENTS
The author wishes to thank Professor R. Tijdeman and Dr. F. Beukers for their helpful remarks. He was supported by the Netherlands Foundation for Mathematics (SMC) with
366 B. M. M. DE WEGER
linancial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO). All machine computations were performed on the IBM-3083 computer at the Centraal Reken Instituut of the University of Leiden.
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DIOPHANTINE EQUATIONS 367
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