A R K I V F O R M A T E M A T I K B a n d 4 n r 17
Communicated 14 September 1960 by T. N A a E ~
On a class o f exponential equations
B y J . W . S. CASSELS
In this note I shall prove the following theorem.
Theorem. Let II1, H2 be two finite sets of rational prime numbers and let P1, P2 be the sets of positive integers all of whose prime factors are in H1, H2 respectively. Then for any fixed integer C :~ 0 there are only a finite number of solutions o/
X - Y = C , X e P 1 , Y e P 2 . (1)
These can all be determined in a finite number o/steps.
The novelty of the theorem lies only in the last sentence. Without it, the theo- rem is a well-known consequence of the theorem of Thue about the approximation of algebraic numbers b y rationals which was subsequently improved b y Siegel and Roth. We shall use instead a result (Lemma 1, below), given by Gelfond [2], which is related to Mahler's p-adie analogue of the Gelfond-Schneider theorem about the transcendence of ~ , where ~ and fl are algebraic.
For the earlier history of the problem solved by our theorem we refer to Nagc!l [1] w 1. Nagell's formulation is different from ours, but the two formulations are readily seen to be identical.
The result which we require is given on page 157 of Gelfond's book, and m a y be formulated for our purposes as follows.
Lemma 1. Let a, b be elements o / s o m e algebraic number field ~(. Suppose that
a u = b v (2)
with rational integers u, v implie~ that u = v = O. Let p be a prime ideal o / ~ for which a and b are p-adiv units. T h e n there is a number x o = x o (a, b, p), which can be deter- mined in a finite number of steps, with the following property:
For any x >1 x o the congruence
a U ~ b ~ (rood pro) (3)
is insoluble in rational integers u, v, m with
We shall need to supplement this b y the trivial Lemma 2.
(4)
231
J. W . S. C A S S E L S . ( ) n ,I
chlss of exponential equations
Lemma 2. L(t ~. b be c h m e n t s o/somr algeb,'aie n u m b e r / i ( h t 7g. Suppos~ that b = 1
~t,',.d that thcr~ i~ a t~lir o/ coprim*: inb!lers U. I such that
C - b v. (5)
L~t t' be ,,t pri,,~e ideal of 3( for which a and b are p.adic units. Then there exists a eo~st,.znt d) = d) ia, ~, pl, u'hi',h can be determined in a finite number o / s t e p s , with the f,')!l,)~cing prop~ rty:
I i the i~t~!ler,~ :t, ,:, m are such that
a'~=b : ( m o d p '~)
then u = t U. c = t V
i,.)r some rational integer t.
(6) (7)
iS)
We sketch the simple proof. F r o m (5) and (6) it follows t h a t
b~=---I (mod p~), (9)
where w = V u - U r .
If ~7) is true. we have
1"1.<2
m a x ([U[, [ V [ ) m a x
([u[, It, l)
~<2 m a x ( t U I - [ V [ ) e x p ( - ~ b + ~ - ' m ) . (10) Here 2
m:,x (1 u [, Iv I)
is fixed. The t h e o r y of the p-adie logarithm now shows immediately t h a t (9) a n d (10) together imply u ' = 0 , provided t h a t ~b is chosen lar~,e enough.We are now in a position to prove the following lemma, of which the theo- rem is an easy consequence.
L e m m a 3. Let II be a finite set o~ rational p r i m e s and let P be the set o / p o s i t i v e in- tegers all o/ whose p r i m e / a c t o r s are in H. L e t D > 0 and E #-0 be rational integers and suppose that no p r i m e /actors of E is in H . T h e n there are emly a finite number o/
solutions Z. Y ot. the equation
Z " - D Y * - = E , ( l l )
where Z is a rational integer and Y e P . These can all be obtained i n a finite number of ste W .
W h e n D is a perfect square the lemma is trivial, so we m a y assume without loss of g e n e r a h t y t h a t the field J f = k ( D i) generated b y D t over the rational field is a real quadratic field. Let q > 1 be the f u n d a m e n t a l unit of k ( D t ) . 3:hen (11) imphes t h a t
Z = Y D t = x ~ ", g-YDi=~t'(tl')" (12)
for some rational integer n, where 0t is one of a finite set of integers of h ( D 89
and where~t', 1/'
are the conjugates of ~t, ~/ respeet4vely. Henceate" - r (r/')" = 2 Y D t . (13)
232
ARKIV FOR MATEM_kTIK, B d 4 n r 1 7
B y (12) t h e r e is a c o n s t a n t
O=O(~,D)>O
such t h a tY>O~ ~. (14)
Since
Y EP,
we h a v eY = 1-I p ~ (15)
l<~a<~S
for s o m e integers ao>~ 0, where Pl, "", P~ a r e t h e p r i m e s in t h e set II. Hence, b y (14)
m a x a~ ~> y~ n,
where ~ = ~ o ( a , D, 1 ] ) > 0 . ~ ' ~ m a y suppose, w i t h o u t loss of g e n e r a l i t y , t h a t
a I > y,n. (16)
L e t p be a p r i m e ideal divisor of Pl in k(D 89 Since E = N o r m ~ r a n d
p I 4 E ,
b y h y p o t h e s i s , i t follows t h a t r162 an d a ' a r e integers for p. Hence, b y (13), (15) a n d (16) we h a v ea----b n
(mod p ~ ) , (17)a = ~ / c x ' , b = r/'/~/
where
a r e integers for p.
B u t n o w
y) n > log 7 n
a n d ~0 n > 4 2 + ~b log n
for all sufficiently large n. H e n c e L e m m a 3 follows from L e m m a 2 or L e m m a 1 a c c o r d i n g as a is a n o r d i n a r y u n i t or not.
W e can n o w d e d u c e t h e t h e o r e m in a few lines. A f t e r d i v i d i n g X , Y, C in (1) be t h e i r c o m m o n divisor, we m a y s u p p o s e w i t h o u t loss of g e n e r a l i t y t h a t X , Y, C are c o p r i m e in pairg. H e n c e we m a y suppose t h a t t h e two sets II1, II~ of p r i m e s a r e d i s j o i n t a n d t h a t C is n o t d i v i s i b l e b y a n y p r i m e in I I I o r Ha.
I n (1) we m a y w r i t e
X = A X ~ , Y = B Y ~ ,
where A a n d B are coprime. T h e r e a r e o n l y a finite n u m b e r of possible v a l u e s for A a n d B. C l e a r l y
Xl e P 1 Yl e P2.
B u t n o w
Z ~ - D Y12 = E ,
where Z = A X 1 , D = A B , E = A C .
W e can now a p p l y L e m m a 3 w i t h [ I = II~. N o t e t h a t we d o n o t use t h e f a c t t h a t X 1 E I I I.
Trinity College, Cambridge, England.
R E F E R E N C E S
1. NAQE~L, T., Sur une classe d'6quations exponentielles. Ark. Mat. 3, 54 (1958).
2. rEJIL~0:I~. A. O : TpaHc~eH~eHTHHe H aareSpaHqecKHe qHcaa; Moscow, 1952.
(Added in proof: An English t~anslati(.n has just appeared, published by the Dover Press.)
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T r y c k t den I0 december 1960
Uppsala 1960. Ahnqvist & Wiksells Boktryckeri AB
