THE RATIONAL SOLUTIONS OF THE DIOPHANTINE EQUATION,
y 2 _ X 3 D.
B y J . W . S. C A S S E L S TRINITY COLLEGE, CAMBItIDGE.
1. We study the rational solutions X, Y of the equation
y 2 ~_ X 3 D (1)
where D is a given integer, a problem of a type considered by Bachet over three centuries ago. When D -: :k I Euler 1 [10] showed t h a t the only solutions are X -- 2, Y - - T3 and trivial ones with X - - 0 or Y - - 0. Apart from a t r e a t m e n t of the special case when D is a perfect cube by Nagell [29], the first significant advance for m a n y years was made by Fueter [12] who writes the equation as
X 3 - - y 2 ~ _ D ,
assumes t h a t D > 0, and studies factorisation in R ( ~ / ( - - D ) ) . This work 'has been extended by Brunner in a doctorate thesis [3]. The case D < 0 was considered by Mordell [26] and then by Chang Kuo-Lung [5].
2. The integral solution ~, ~], ~ of the equation
~ - ~ --~ A~ ~ , (~ ~ 0 )
where A is a given integer, is trivially equivalent to the rational solution of (1) with D ~- 243aA 2 by putting
X : Y : A = 223~:2232(~--~1):~-~rt.
The case A = 1 is, of course, F e r m a t ' s problem with exponent 3. The equation with general A was extensively investigated in the 19th Centu?y by Lucas [19], Pdpin [36, 37] and Sylvester [40]; and Sylvester states t h a t he either had a s o l u t i o n or
1 F o r references see end of p a p e r .
244 J . W . S . C a s s e l s .
knew the equation to be insoluble for all positive A < 100 except 1 A ~-- 66. Further results have been given b y Hurwitz [17], Faddeev [11] and Holzer [16]. Much of this work is summarized b y Nagell [31]. [added in the proof]. Since this was written new and interesting work has been done b y Dr. E. S. Selmer (so far unpublished).
3. The equation (1) is, of course, a special case of
y 2 = x ~ C X _ D (2)
where C and D are integers. This was studied b y Poincar~ [38] who noted that the values of the parameter u corresponding to rational solutions form an additive group, II, when the usual parametrization Y ~ 1 , ~4) (u), X ~ 4~(u) is employed.
This group was shown to have a finite basis by Mordell [25]. A more precise form of this result was given b y Weil (cf. theorem II) who gives an elementary proof [42]
as well as a deep proof of a far-reaching generalization [41]. In a doctorate thesis, Billing [2] has given a general study of (2) using methods based on Weil's theorem and, in particular, he gives a complete solution of (1) for all IDf < 25 in the sense that he gives a complete basis for U. He does not, however, give a detailed account of the method of obtaining these results. The present work was done in ignorance of Billing's paper; indeed, it was not until a late stage that I realized t h a t the algorithm which I employed was that underlying Weil's theorem II. In it, I have developed a more detailed theory of (1) than is given b y Billing and have given general theorems as well as carrying the solution up to f D! < - 5 0 . With one exception (D ~ - - 1 5 ) , m y results confirm Billing's in the range IDi < 25 studied b y him. I have been led to compute a table 2 of class-numbers and units for all cubic fields R(~/D) with ]D I < 50; which I do not think has been given before, although a number of partial tables exist. Finally, it should be remarked that (2) is the subject of a series of papers b y Nagell [30, 32, 34, 35] and that other aspects of the problem have been studied b y Ch~telet [6, 7], Lutz [20] and Lind [18].
4. In part I, I give a resumd of the general theory of (2) and discuss its relevance to (1) in general terms. In part I I the general discussion is carried further using the specific arithmetical properties of the relevant cubic number-fields, and in part I I I the actual applications s r e made.
1 I n s o l u b l e b y t h e o r e m V I I I . Table 2.
T h e R a t i o n a l S o l u t i o n s of t h e D i o p h a n t i n e E q u a t i o n s y 2 _~ X 3 _ D . 2 4 5
P a r t I.
5. Let (X', Y') and ( X " , Y " ) be any two rational solutions I of y2 = X 3 C X _ D with parameters u', u" and let (X'", Y'") be the solution with parameter u ' " --- u ' q - u " . Then (X', Y'), (X", Y") and (X'", -- Y'") lie on a straight line y -~ A x q - B b y a known result 2, where A and B must be rational. Hence X', X " , X ' " are the roots of
X 3 - - C X - - D - - ( A X q - B ) 2 --- O. (3) The left-hand side of (3) must be identical with ( X - - X ' ) ( X - - X " ) ( X - - X ' " ) and so, if (~ is a root of ~ 3 - - C ~ - - D = O,
( X ' - - 6 ) ( X " - - 6 ) ( X ' " - - 5 ) = (A6q-B) 2 , i . e .
= e ( 4 )
In other words, if squared factors are ignored, the values of X - - 6 form a multi- plicative group homomorphic to 11. There are three groups (~1, (~2, (~s (say) corre- sponding in this w a y to the three values 51, ~2, 6a (say) of ~. If 5 a - - C S - - D is irreduc- ible, the numbers X - - S j (j = 1, 2, 3) are conjugate algebraic numbers, and the three groups (~j run entirely parallel to one another. If, however, 5 3 - - C ~ - - D is reducible, this parallelism does not necessarily hold and so we are compelled to introduce a group (~ in terms of "triplets".
A triplet {aj} is defined as a set of three numbers al, a2, as such that o~j E R(Sj) and the operations of addition and multiplication for triplets are defined b y
= =
Then the set of triplets { X - - d j } is clearly also a multiplicative group N homomorphic to 11, when squared (triplet) factors are ignored. Obviously, when 6 3 - - C S - - D is irreducible (~ is isomorphic to each of the groups gO;..
We denote, further, b y 211 the set of 2u, u E 11. Clearly 211 forms a group. We denote the quotient group of 1I and 211 b y 11/(211). Then the following three theorems hold.
T h e o r e m I (Mordell). The group 1I has a finite basis.
T h e o r e m II (Wail). (~ is isomorphic to 1I/(21I). The element of ~ and the element
I (X, Y) (with or w i t h o u t affixes) will a l w a y s be a r a t i o n a l s o l u t i o n of Y ~ ~ X a - - C X - - D . 2 cf. W h i t t a k e r a n d W a t s o n [44].
246 J . W . S . Cassols.
of tt/(21l) belonging to the same solution of y3 ~ x a _ C X _ D correspond to one another in the isomorphism.
T h e o r e m I I I . Let (I) w be the number of independent generators of infinite order in l~t, (II) g be the number of generators of ~ and (III) s • 0, 1, 2 according as O a CO--D = 0 has no, has one or has three rational solutions. Then w ~ g--s.
Theorem I I I is really a corollary of theorem I I . F o r proofs we refer to t h e paper of Weil [42] or the book of D e l a u n a y a n d F a d d e e v [9].
6. I f (~ is known, then, b y theorems I I a n d I I I , the s t r u c t u r e of 1I is k n o w n , except for its generators of finite order. A t h e o r e m has been given b y L u t z [21]
which, while n o t completely characterizing the solutions of finite order of y 3 = X 3 - - C X - - D (i. e. those solutions whose p a r a m e t e r s are of finite order in H), reduces t h e problem, w h e n C a n d D are given, to t h e s t u d y of a manageable n u m b e r of cases. We shall n o t need it here, b u t quote it for completeness.
T h e o r e m IV (Lutz). I f X , Y is a rational solution of y3 ~ X a _ C X _ D of finite order, then X and Y are integers and Y3/(4Ca--27D3).
This is superseded in t h e case C ~ 0 b y
T h e o r e m V (Fueter-Billing). Solutions of y3 X a - - D with X = 0 or Y ~ 0 are of order 3 and 2 respectively. The only other solutions of finite order are the two
following:-- X -- 2, Y ~ • D -- --1 ; (5)
X = 233, Y ~ 32333 , D ~ 2433; (6)
of order 6 and 3 respectively.
7. A n o t h e r general t h e o r e m is
T h e o r e m VI (Fueter-Billing)i The number w* (say) of independent generators of infinite order of the group lI* (say) of the equation y3 _~ XaA_27D is equal to the corresponding number w for y 2 = Xa D.
The interdependence between these two equations has been k n o w n for a long time. I t is connected with t h e possibility of " c o m p l e x m u l t i p l i c a t i o n " of t h e para- m e t e r u b y 1/(--3).
8. Finally, we enunciate t h e a l m o s t trivial
T h e o r e m V I I . There is a 1--1 correspondence between the rational solutions X , Y of y3 = X3 C X _ D and the integer solutions x, y, t of
T h e R a t i o n a l S o l u t i o n s o f t h e D i o p h a n t i n e E q u a t i o n y 2 _: X a D . 247
y2 ~ x a _ C x t 4 Dt6, t > O, (x, t) _~ (y, t) = 1. (7) This follows immediately by putting X ~ - x ~ r , Y ~ y / s where x, y, r, s are integers and the fractions are in their lowest terms. Comparison of denominators on both sides of y 2 =_ X a C X _ D g i v e s s2 _= r 3 and hence s ~ t 3, r --- t 2 for some t.
I t will be more convenient to use this form in future. We note t h a t the multiplicative group (~ of the triplets {X--~j} m a y also be defined as the grou~ of the {x--t2(Sj}, squared factors again being ignored.
P a r t II.
9. We shall now confine ourselves to the equation
y 2 = x 3 D t ~ , t 4 = O , (x,t)~-- ( y , t ) : 1, (8) where x, y, t are integers. We m a y clearly assume t h a t D is sixth-power-free and, by theorem VI t h a t 27r Any such D can be put in the form 1
D : E F 2 G a, E > O, F > O, 3 ~G , (9)
(E, F) = 1 , (10)
E , F , G squarefree. (11)
By theorems I I I and V the structure of the group 1I of solutions can be found from t h a t of the group (~. Until further notice, we shall assume t h a t D is not a perfect cube and so, by theorem I I I , the number of independent generators of infinite order of 11 is equal to the number of generators of ~ . Further, all three expressions x - - t 2 6 j (j = 1, 2, 3) where ~j runs through the roots of 63 = D, are conjugate, and so we need study only one, say x--t26 where 5 is the real cube root of D. We first state some properties of the cubic fields R(6). For proofs see a paper by Dedekind [8]
or the general theory in Weyl [43].
10. Write
(~ -~ GA, D * -~ E F " ~- A3 > 1 .
Small Greek letters denote numbers in R ( A ) , Gothic small letters denote ideals, as also do square brackets enclosing a (possibly redundant) basis. If D * ~ie :~ 1 rood 9 integers in R(A) have the basis {1, A, ~ 2/F} b u t if D* ~ -~-1 rood 9 the basis is {1, A, A 2 / F , 8 9 In particular
1 F o r l e t pS]]D w h e r e p i s a r a t i o n a l p r i m e . A c c o r d i n g a s s = 1, 2, 3, 4, 5 w e m a k e
(s = 1) PilE, p~FG; (s = 2) p]]F, p~EG; (s = 3) p]/G, p~EF; (s = 4) PilE, pfF, p]]G;
(s = 5) p~E, p]]F, p//(~. F u r t h e r , s i g n G = s i g n D .
248 J . W . S . Cassels.
L e m m a 1. The number a + b A , where a and b are rational, is an integer in R ( J ) if and only i f a and b are integers.
T h e r a t i o n a l p r i m e s p f a c t o r i s e as follows in R ( J ) :
p/D*: p - = p~ w h e r e p ~-- [p, J ] or p --- [p, JJ/F] a c c o r d i n g as p i E or p/F.
p ~ 3 ~ D * , D * ~!-: :}=1 rood 9 : 3 =- ~3, 1: ~ [3, T z l ] w h e r e D * _~_ =kl m o d 3.
p : 3 / D * , D * -~ ! l rood 9 : 3 =- 1:2~ w h e r e -- [3, 1 T J , ~ ( l I J §
= [3, ~=~J, ~ ( - 2 + + J + J ~ ) ] . H e r e r~ = [3, 1 ~= J ] .
p : = - - 1 m o d 3, p / D * : p = pq w h e r e
= [p, d - - A ] , q = [p, d J + d J + A ~]
a n d d is t h e u n i q u e r o o t of t h e c o n g r u e n c e d a ~ D * m o d p.
p a n d q are of t h e first a n d s e c o n d d e g r e e s r e s p e c t i v e l y . ~
p ~ 1 rood 3,
p/D*,
D* a cubic r e s i d u e of p : p ~ P~PJPa w h e r e pj --- [p, d j - - J ] a n d d~, dj, d3 a r e t h e d i s t i n c t r o o t s of d a ~ D* m o d p.p ~ 1 rood 3,
p/D*,
D* n o t a cubic r e s i d u e of p : p r e m a i n s p r i m e in R ( J ) . T h e following a r e q u o t e d f o r r e f e r e n c e as l e m m a s . T h e p r o o f s a r e e l e m e n t a r y . L e m m a 2. Let a, b be rationa$ integers and p / ( a , b) a rational prime. Suppose p ~ 3 if D* ~ • mod 9. Then either p is prime to a § or[p, a + b J ] =- pw where p is some first degree prime divisor of p.
L e m m a 3. I f 0 is a prime ideal of the first degree the rational integers O, 1, 2 , . . . , p - - 1 are a complete set of incongruent residues rood p where p -~ N o r m p.
L e m m a 4. I f q is a prime ideal of the second degree the numbers a § ( a , b = 0, 1 , 2 . . . p - - l )
are a complete set of incongruent residues m o d q where p2 ~ N o r m q.
L e m m a 5. Let t be a prime ideal and t~/2 for some s > O. Then ~ ~ fl rood i ~
1 i e N o r m p ~ p , N o r m q = p J .
T h e R a t i o n a l S o l u t i o n s o f t h e D i o p h a n t i n e E q u a t i o n y 2 = X a _ D . 249 implies a 2 ~ f12 m o d t ~. Indeed it implies a 2 ~ f12 m o d t ~+1 i f t is
of
the first degree.I n particular a 2 ~_~ 1 m o d t s i f t is of the first degree and t ~ a.
11. T h e u n i t s of R(z]) are all of t h e f o r m ~:e n w h e r e s > 0 is t h e f u n d a m e n t a l unit. F o r o u r p r e s e n t p u r p o s e it is e n o u g h t o k n o w a n y o d d p o w e r ~ ~ -~ e ~n+~ > 0 of s. Values of ~ a n d of t h e c l a s s n u m b e r h of R(A) are g i v e n in t a b l e 2. As a special case of a k n o w n general t h e o r e m we h a v e 2
L e m m a b. I f h is odd, ~ is not a quadratic residue of 4.
L e t n o w ~7~be t h e p r i n c i p a l class of ideals in R(A) a n d gl, g2 . . . . , t~k t h e classes of o r d e r 2, if any, i.e. ~. 4= ~ 8 2 -~ ~ . L e t c j e ~ a n d choose ?~- > 0 such t h a t [?j] -~ c~. T h e n e v e r y n u m b e r 0 > 0 of R(A) for which [0] is t h e s q u a r e of an ideal has one of t h e f o r m s
T h e yj can be chosen t o be integers p r i m e t o a n y g i v e n ideal m. H e n c e , if 0 is an integer, ~ c o n t a i n s in its d e n o m i n a t o r o n l y ideals p r i m e to m. I n p a r t i c u l a r , we shall suppose h e n c e f o r t h t h a t t h e ~j are p r i m e t o 2. T h e r e is a g e n e r a l i s a t i o n of l e m m a 5 which s t a t e s t h a t t h e n precisely half of t h e n u m b e r s
1, V, ~j, W~- (j = 1, 2 . . . . , k)
are q u a d r a t i c residues of 4. W e shall n o t use this g e n e r a l i s a t i o n in t h e general t h e o r y b u t in n u m e r i c a l w o r k we shall use t h e c o r o l l a r y t h a t if V is n o t a q u a d r a t i c residue of 4 t h e ~ / m a y be chosen all t o be q u a d r a t i c residues, t o give a n o r m a l i s a t i o n of t h e ~j w h e r e possible. This is a l w a y s possible in t h e r a n g e ID[ < 50, since t h e r e is n o t a q u a d r a t i c residue of 4.
12. Consider t h e f a c t o r i s a t i o n
y2 = ( x - - Gt2z])(x2 +Gt~xA + G~t4A 2).
1 I t is d i f f i c u l t to decide if a g i v e n u n i t ~1 ~ 0 is a f u n d a m e n t a l u n i t b u t e a s y to decide if it is a p e r f e c t s q u a r e . I f n o t , p u t ~] ~ ~1.
2 L e m m a 6 b e l o n g s to t h e g e n e r a l t h e o r y of class-fields as e x p o u n d e d b y H a s s e [13, 14]. T h e full force of t h i s t h e o r y is n o t r e q u i r e d a n d it is p o s s i b l e to b a s e a p r o o f o n t h e s i m p l e r t h e o r y of r e l a t i v e - q u a d r a t i c fields d e v e l o p e d b y H i l b e r t [15]. B y S a t z 4 o n p a g e 374 of [15] if ~ is a q u a d r a t i c r e s i d u e of 4, t h e r e l a t i v e - d i s c r i m i n a n t of R(z~, ~/~) o v e r R(A) is u n i t y , s i n c e it h a s n o p r i m e f a c t o r s . B y S a t z 94 o n p a g e 155 ( a n d t h e r e m a r k o n p a g e 156 e x t e n d i n g it t o 1 ~ 2 in t h e t h e r e n o t a t i o n if a f u r t h e r c o n d i t i o n is satisfied) it follows t h a t h is e v e n .
I n t h e l a n g u a g e of class-field t h e o r y if ~ is a q u a d r a t i c r e s i d u e of 4 t h e field R ( A , ~/~) is u n r a m i f i e d (unverzweigt) o v e r R(A) a n d so is t h e class-field to s o m e a b s o l u t e ideal g r o u p of i n d e x 2. T h i s i m p l i e s t h a t h is e v e n .
250 J . W . S . Cassels.
A n y c o m m o n divisor a of t h e two t e r m s on t h e r i g h t h a n d side m u s t divide x2-~ Gt2xA ~-Get4A ~ _ (x--Gt2z~ )~ = 3Gt~xA
a n d so a / 3 G A since (x, t) = 1. H e n c e [ x - - G t 2 A ] -= ab 2 for some ideal b a n d a / 3 G A . W e classify t h e p r i m e divisors of ~ as follows
(i) p / d . T h e n p a _ [p] where p = N o r m p . Since p occurs in y e = N o r m ( x - - G t 2 d ) t o an e v e n power, p occurs in x - - G t 2 d to an e v e n power, a n d so m a y be a b s o r b e d in b.
(ii) q / G b u t q f A , so q f 3 b y (9). T h e n q / x since q / G a n d q / ( x - - G t 2 A ) i. e. q / x where q is t h e r a t i o n a l p r i m e d i v i s i b l e b y q. W r i t e x = qxl, G -= qG 1. B y l e m m a 2, e i t h e r x l - - G ~ t 2 A is p r i m e to q or [q, x~--Glt~A] --~ q,W for some first degree p r i m e divisor q' of q, since qT~Glt 2 [as q ~ G 1 b y ( l l ) a n d q ~ t b y (8) (and q/x)]. B u t y~ =- N o r m ( x - - G t 2 A ) = q 3 N o r m ( x ~ - - G l t 2 A ) a n d so t h e second a l t e r n a t i v e holds a n d q' divides x ~ - G l t 2 A to a n o d d power.
(iii) p/3 b u t p f G z l . H e n c e 3/~D b u t 3 / y since y ~ = N o r m ( x - - G t 2 A ) . Conse- q u e n t l y x a ~ GateD * m o d 9 so this case occurs o n l y w h e n D* ~ • 1 m o d 9 i. e.
w h e n [3] = r2~. N o w x 3 ~ G3t6D * ~- ! G 3 t e m o d 9 implies x ~ • 2 m o d 3 a n d so r ~ / ( x - - G t 2 ~ ) since r~ = [3, 1-TA]. B u t 3 ~ ( x - - G t 2 ~ ) (lemma 1) so r / / ( x - - G t 2 ~ ) . H e n c e ~ occurs in x - - G t 2 A t o an o d d p o w e r since y 2 = N o r m ( x - - G t 2 A ) a n d N o r m ~ = N o r m ~ = 3.
All this p r o v e s
L e m m a 7. L e t ql,. . ., ql be the rational p r i m e s such that qj/(x, G) but q j t D * . T h e n
[ x _ G t 2 A ] = q~ . . . q_L. b2 ' (12)
q~ qz
where qj is some f i r s t degree p r i m e divisor o f qj a n d b is some ideal; except that, w h e n D * ~ i i rood 9 a n d 3/y,
[ x _ G t 2 A ] _= q~ . . . q_t. ~ . b2 (13)
q~ q~
I n p a r t i c u l a r [ x - - G t 2 A ] = ab 2 where a is one of a finite set. H e n c e ~ T f : ~7/~ 2 w h e r e ~ a n d ~ are t h e ideal classes to which a a n d b belong a n d ~7[(as b e f o r e ) i s t h e principal class. F o r g i v e n a a n d so for g i v e n ~71, this e q u a t i o n for ~ m a y be insoluble
T h e R a t i o n a l S o l u t i o n s of t h e D i o p h a n t i n e E q u a t i o n y 2 = X a D . 2 5 1
(e. g. when 1 the class-number h ~- 2 and ~ 4: gT(). If however ~/~2 = ~7( has the solution ~--- ~ , , the other solutions are all the ~ = ~16" where 6" (as before) runs t h r o u g h M1 the solutions of b ~" = ~ , t? 4: ~ . We m a y choose bl E ~ , and prime to a n y given ideal m. Then ab~ is a principal ideal, say ab~ = [~t]. Now [ x - - G t 2 ~ ] = ab ~ : [ab~][bb~-l] 2 where bb~-lE ~j~-11 : ~. for some j, or : ~Tf.
Hence finally
x _ G t 2 A : •162 or • or + 7 j ~ ~ or • say
x _ G t 2 A : / ~ 2 (14)
where # is an integer in R ( A ) taken from a finite set, which m a y be chosen so t h a t a, though not necessarily an integer, has in its denominator only factors prime to a n y given 2 m. We note t h a t # > 0 since xa--(Gt2A) 3 = y2 > O.
We shall say t h a t two values of # are essentially s i m i l a r if their quotient is a square in R(A), otherwise essentially dissimilar. The values of # which actually correspond to solutions of (8) form the multiplicative group | when squared factors arc ignored, which we discussed earlier. We shall use this group-property frequently.
We note in particular t h a t we can assume t h a t # is essentially distinct from 1 when investigating the number of generators of (~. As we remarked earlier, the number of generators of (~ is the number of generators of infinite order of lt.
13. All the argument so far applies equally to the equation y2 : x3_~Dt 6 in which the sign of D is changed and leads to the equation
x + G t 2 A : #~2 (15)
in which # has the same a priori a possible values as for (14). I t will often be con- venient to discuss (14) and (15) simultaneously and then, in numerical work, we shall always mean by D, ~ the positive values and make the appropriate changes in the formulae when discussing negative D. General theorems will, however, apply equally to either positive or negative D.
1 A n u m e r i c a l case is D = ~:88 a n d 2Ix. T h e n G = 2, D* = l l , h = 2 a n d ~t = II is t h e second-
3 (
degree divisor of [2] in R( U 11), w h i c h is n o t a p r i n c i p a l ideal. H e n c e t h e r e are no s o l u t i o n s w i t h D = -t- 88, 2Ix [indeed n o n e w i t h D = + 88 a t all]. A similar a r g u m e n t in a q u a d r a t i c field h a s b e e n u s e d b y Mordell a n d i n d e p e n d e n t l y b y Marshall H a l l ( b o t h u n p u b l i s h e d ) .
2 W e r e q u i r e o n l y 111 ~ 6.
a i. e. so far as t h e discussion in p a r t I I is concerned. W e shall use a priori in t h i s sense t h r o u g h o u t p a r t I I I .
2 5 2 J . W . S . C a s s e l s .
P a r t I I I .
14. I n this p a r t w e give a n u m b e r of general t h e o r e m s covering m o s t o f t h e values of D such t h a t IDI < 50 a n d t h e n dispose of t h e rest individually.
15. D odd. W e e x a m i n e (14) m o d u l o powers of f a n d H, t h e p r i m e divisors of [2] of t h e first a n d second degrees respectively. W e n o t e t h a t
f = [2, 1-~-6], [ 2 = [4, D - - 6 ] , f3 = [8, n - - 6 ] ,
u = [2, 1--kS-kS2], u 2 = [4, 1--kDSq-52], u 3 = [8, 1 - k D O - k 5 2 ] . W e p r o v e first
T h e o r e m V I I I . I f D is odd a n d (14) is true then either/~ is a quadratic residue o f 4 or
~ D - - 5 , 2 - - 5 , D - - 2 6 rood u 2 . (16) One a n d o n l y one of x, y, t is e v e n since (x, t) = (y, t) = 1 a n d we t a k e t h e possibilities in t u r n .
(i) t even. T h e n x ~ x 3 ~ y 2 ~ 1 m o d 8 so / ~ 2 = x _ t 2 6 ~ 1 m o d 4 a n d /~
is a q u a d r a t i c residue of 4.
(ii) y even. T h e n x ~ x 3 ~ D t ~ ~ D m o d 4 a n d s o / ~ 2 = x _ t 2 6 ~ D - - 6 m o d 4.
B u t ~ is p r i m e to u since x - - t 2 5 is ( b y l e m m a 2). H e n c e t~ = ( x - - t 2 6 ) ( 1 / ~ ) 3 ~ ( D - - 6 ) ( 1 / ~ ) 2 m o d u 2 .
B y l e m m a 4, ( l / s ) ~ 1, 6, 1 + 6 m o d u so, b y l e m m a 5, ( 1 / ~ ) 2 ~ 13 , ~2, ( 1 + 5 ) 2 m o d u 2 a n d t h e n (16) holds.
(iii) x even. T h e n D ~ Dt~ ~ _ y 2 ~ - - 1 m o d 8. I f 4/x, t h e n / ~ 3 = x - - t 3 5 ~ --5 ~ (53) 3 m o d 4 so/~ is a q u a d r a t i c residue of 4. If, h o w e v e r , 2]]x t h e n
/ ~ 2 = x _ t 2 6 ~ 2 - - 5 m o d 4 a n d so (16) holds.
This concludes t h e proof.
C o r o l l a r y 1. I f D is odd a n d / ~ is not a quadratic residue o f 4 at least one o f (14) or (15) is insoluble.
F o r if (16) is t r u e t h e c o r r e s p o n d i n g congruences in which t h e signs of D a n d are s i m u l t a n e o u s l y c h a n g e d c a n n o t be t r u e .
I n p a r t i c u l a r ,
C o r o l l a r y 2. I f D -j~ ~: 1 m o d 9 is odd a n d cube-free 1 a n d i f h D is odd, n o n - 1 I t is e n o u g h t h a t t h e r e is n o p r i m e p w i t h pS//D.
T h e R a t i o n a l S o l u t i o n s of t h e D i o p h a n t i n e E q u a t i o n y 2 ~ X a _ D . 2 5 3
trivial solutions do not exist for both y2 ~ x a • I f non-trivial solutions do exist for one of these equations then the corresponding group 1I has precisely one generator of infinite order.
F o r u n d e r t h e conditions of c o r o l l a r y 2 t h e o n l y a priori possible v a l u e of # essentially dissimilar f r o m 1 is # : ~, which is n o t a q u a d r a t i c residue of 4 b y l e m m a 6.
H e n c e (~ has a t m o s t one g e n e r a t o r , i. e. 11 has at m o s t one g e n e r a t o r of infinite o r d e r ( t h e o r e m I I I ) and, b y corollary 1, such a g e n e r a t o r exists for a t m o s t one of t h e t w o equations. F i n a l l y , b y t h e o r e m V, t h e o n l y solutions of finite o r d e r for t h e D u n d e r c o n s i d e r a t i o n are trivial.
W e n o w consider some n u m e r i c a l examples.
N o n - t r i v i a l solutions a c t u a l l y exist (table 1) for t h e following values of D for which h D is odd (table 2) a n d which satisfy t h e o t h e r conditions of corollary 2:
D ~-- - - 3 , --5, ~-7, - - 9 , ~ 1 3 , -~21, ~ 2 3 , ~ 2 5 , ~-29, - - 3 1 , - - 3 3 , - - 4 1 , -t-45, ~ 4 9 . H e n c e 1I has one g e n e r a t o r of infinite o r d e r for these D whereas t h e r e are no non- trivial solutions a t all for
D ~ ~ 3 , -k5, - - 7 , + 9 , etc.
Consider n o w D -[- • 1 m o d 9 a n d cube-free, b u t w i t h h D even. T h e n t h e o n l y possible values of/~ essentially d i f f e r e n t f r o m 1 are /~ ~-- ~, ~j, ~l~j. Suppose f u r t h e r t h a t tDI < 50. T h e n (table 2) t h e g r o u p of ideal-classes is cyclic, so essentially o n l y one v a l u e of ~j : ~ exists~ a n d ~ is a q u a d r a t i c residue of 4 b u t ~ a n d ~ are not.
As t h e values of/~ for which (14) is soluble f o r m t h e m u l t i p l i c a t i v e g r o u p (~, t h e g r o u p U has no, one or t w o g e n e r a t o r s of infinite o r d e r a c c o r d i n g as none, o r one or all t h r e e possible v a l u e s of # essentially dissimilar f r o m 1 do in f a c t h a v e solutions in (14).
I n p a r t i c u l a r , consider D ~- q - l l . N e i t h e r of t h e solutions (x, y, t) --~ (3, 4, 1) or (15, 58, 1) l e a d s to a v a l u e of # which is a q u a d r a t i c residue of 4 (as in t h e p r o o f of t h e o r e m V I I I ) n o r do t h e y belong t o t h e same # since t h e r a t i o
(3-~)/(15-~)
is n o t a p e r f e c t s q u a r e 1. H e n c e solutions exist b o t h f o r / ~ ~ ~/ a n d for # ~ V~, so also for ~ ---- ~ b y t h e g r o u p - p r o p e r t y of #. T h e c o r r e s p o n d i n g g r o u p 1111 has t h u s t w o g e n e r a t o r s of infinite order. B y c o r o l l a r y 1 t h e r e are t h e n no solutions for D : -- 11 e i t h e r w i t h /~ ~--V or # ~--V~, so w h e n D - ~ --11 t h e o n l y possible v a l u e for /~
essentially dissimilar f r o m 1 is /~ ~ ~ i. e. t h e g r o u p 11-11 can h a v e a t m o s t one
1 I t is not a quadratic residue rood [5, (~--1], the first-degree prime divisor of 5.
254 J . W . S . Cassels.
g e n e r a t o r of infinite order. I t h a s p r e c i s e l y one s u c h g e n e r a t o r since n o n - t r i v i a l s o l u t i o n s do in f a c t exist. H o w e v e r / ~ = 1 o r / ~ = ~+ is a q u a d r a t i c r e s i d u e of 4 f o r all t h e s e solutions a n d h e n c e so is I x~-t26. Since - - 1 1 : : - - 1 rood 8, t h i s implies 2/t, as in t h e p r o o f of t h e o r e m V I I I . I n p a r t i c u l a r X = x/t 2 a n d Y = y/t 3 c a n n o t be integers. T h e s o l u t i o n g i v e n in t a b l e 1 for D ~- - - l l , n a m e l y (x, y, t) = ( - - 7 , 19, 2) c o r r e s p o n d s t o ~ x~-t"5 = ~+~2 a n d n o t to x~-t25 = ~2 since - - 7 ~ - 4 6 =~ 5 m o d t a b u t a s ~ I m o d t 3 b y l e m m a 5.
S i m i l a r a r g u m e n t s a p p l y to D = ~ 3 9 , --+43, - + 4 7 . T h e o n l y m o d i f i c a t i o n n e c e s s a r y w h e n D = -+ 15 is t h a t a l t h o u g h we c a n p r o v e t h a t , w h e n D = ~-15, x - - t 2 b is a q u a d r a t i c residue of 4 t h i s does not i m p l y 2/t since 15 _= - - 1 m o d 8 a n d we m a y h a v e 4/x as in t h e case (iii) of t h e p r o o f of t h e t h e o r e m . T h i s in f a c t does occur.
W e l e a v e f o r l a t e r c o n s i d e r a t i o n t h e cases D _= -+ 1 m o d 9.
16. W e m a y s t r e n g t h e n t h e o r e m V I I I s o m e w h a t b y considering c o n g r u e n c e s to p o w e r s of t as well as to p o w e r s of u. W e p r o v e t h e s t r e n g t h e n e d f o r m a l t h o u g h it is n o t r e q u i r e d to deal w i t h ID] < 50. W e r e q u i r e first
L e m m a 8. I f D is odd a n d (14) is true but/~ is not a quadratic residue of 4 then either 2/y or 2//x, the second case occurring only when D ~ - - 1 mod, 8. Further/~ ~-~ 3 m o d ~2 and ~ ~ 7 or ~ 3 m o d ia, in the first case according as 2//y or 4/y, and i n the second case according as x ~ - - 2 or x --= -+ 2 m o d 8.
T h e p r o o f of t h e first s e n t e n c e h a s a l r e a d y b e e n g i v e n in t h e p r o o f of t h e o r e m V I I I . W e r e c o n s i d e r cases (ii) a n d (iii) of t h a t proof.
(ii) 2/y. S u p p o s e 2"//y. T h e n t~"//(x--t25) since y 2 : N o r m (x--t2~) a n d u/~(x--t25) b y l e m m a 2. H e n c e
y2 = xa Dt6 = / ~ 3 ~ 6 ~ _ 3 / ~ x 4 t 2 ~ . . ~ 3 / ~ X 2 t 4 ~ 2
on s u b s t i t u t i n g for x in t e r m s o f / ~ a n d ~ a n d so
y )2 3~2/~ /~2~4
= - - ) 2 T " ( ! 7 )
I f a > 1 t h e second a n d t h i r d t e r m s on t h e r i g h t a r e divisible b y t 4 a n d t h e first, b y l e m m a 5, is c o n g r u e n t t o 1 m o d t 3. H e n c e 3/~ ~ 1, # _~ 3 m o d t 3. I f h o w e v e r
i ~ ~ ~ 1 1 i n a c c o r d a n c e w i t h t h e c o n v e n t i o n i n t r o d u c e d a t t h e e n d o f p a r t I I .
T h e R a t i o n a l S o l u t i o n s of t h e D i o p h a n t i n e E q u a t i o n Y ~ = X a - - D . 2 5 5 a = 1, t h e t h i r d t e r m is still divisible b y t 4 b u t t h e s e c o n d o n l y b y t 2 a n d so ~ is c o n g r u e n t t o 4 r o o d i a. H e n c e 3/, ~ 5, /~ ~ 7 r o o d t a.
(iii) 2/x so D ~ - - 1 r o o d 8. H e r e ~ is p r i m e t o t since y is t o 2 a n d so
# ~- tta 2 = x - - t 2 6 =~ x - - 6 ~_ x + l m o d t a .
Since w e h a v e a l r e a d y s h o w n t h a t 2 2 / x if:/, is n o t a q u a d r a t i c r e s i d u e of 4 t h i s c o n c l u d e s t h e p r o o f of t h e l e m m a .
I f # is n o t a q u a d r a t i c r e s i d u e of 4 we m a y n o w w r i t e
# ~ 3 - ~ 4 k m o d t a , w h e r e k ~ 0 or 1, a n d p r o c e e d t o p r o v e
T h e o r e m I X . T h e o r e m V I I I r e m a i n s true i f (16) is replaced by
# ~ D - - 6 ~ 4 k + 4 / ( l + 6 ) , - - 2 D - - 6 ~ - 4 k - ~ 4 1 6 , 2 - - D - - 2 6 + 4 k 6 + 4 1 m o d u 3 , (18) where k has j u s t been d e f i n e d a n d 1 m a y take both values 0 or 1.
I t is e a s i l y v e r i f i e d t h a t o n l y h a l f t h e n u m b e r s w h i c h a r e q u a d r a t i c r e s i d u e s o f u 2 a r e q u a d r a t i c r e s i d u e s of u a. T h u s if fl ~ 1 r o o d u t h e n fl~ ~_ 1, 5 m o d u a b u t
1-{-4~ a n d 5 + 46 a r e n o t q u a d r a t i c r e s i d u e s of u ~. T o p r o v e t h e o r e m I X w e n e e d n o w o n l y r e c o n s i d e r cases (ii) a n d (iii) of t h e o r e m V I I I a n d use l e m m a 8.
(ii) 2/y. H e r e
x ~ x a ~- y 2 + D t ~ ~- 4 k N D m o d 8 , b y l e m m a 8. H e n c e
# = (x--t26)(1/o~) 2 ~_ ( D N 4 k - - 6 ) ( 1 / o ~ ) 2 m o d u a a n d so (18) h o l d s .
(iii) 2/x. T h i s m a y be s i m i l a r l y d e a l t w i t h .
17. 2 / / D . W e c o n s i d e r c o n g r u e n c e s t o p o w e r s of t w h e r e t - - [2, 6], t ~ = [2, 65], t 3 = [2]
T h e o r e m X . I f 2 / / D a n d (14) is soluble then either # is a quadratic residue o f 4 or /~ ~ 1 • , 1 + D - - 6 , l - - D + 6 r o o d t 7 --- 4[2, 6 ] . (19) I f e i t h e r x or y w e r e e v e n t h e n so w o u l d t h e o t h e r be, a n d t h e n D t ~ -~ x a - - y ~ ~ 0 r o o d 4 i. e. 2/t c o n t r a r y t o (x, t) = (y, t) ~ 1. H e n c e 2 ~xy. T h e r e a r e t w o cases (i) t e v e n a n d (ii) t o d d .
1 If tf fll a n d tf/32 t h e n / 3 1 ~ / 3 2 raod t since N o r m t = 2; similarly if t'~11/31 a n d ts/I/32 t h e n / 3 1 ~ / 3 . , rood t s+l.
256 J . W . S . Cassels.
(i) t even. H e r e x ~ x 3 _= y~ ~ 1 m o d 4, s o / z ~ 2 = x--t2~ ~ 1 m o d 4 a n d ~u is a q u a d r a t i c r e s i d u e of 4.
(ii) t odd. T h e n x ~ x a - ~ y 2 + D t 6 : ~ I + D rood 8 so I~o~ = x--t25 ::_ I + D - - ~ m o d [8] ---- f g .
B u t y is odd, so ~ is p r i m e to f a n d t h e n ( l / a ) ~ 1, 1 + ~ , 1 + 5 3 , 1 + ~ + 5 2 m o d [ 2 ] - - - - t 3. H e n c e , b y l e m m a 4,
(1/~) 2 ~ 1, 1 + 2 ~ + ~ 2, 1 + 2 ~ + 2 5 2 , 5 + 3 5 ~ m o d t 7 , (20) a n d /z ---- (x--t25)(1/oO 2 satisfies (19).
T h i s c o n c l u d e s t h e p r o o f . W e n o t e t h a t t h e r i g h t h a n d side of (19) r e m a i n s u n a l t e r e d w h e n - - D , - - ~ a r e s u b s t i t u t e d f o r + D , + ~ r e s p e c t i v e l y .
L e t us n o w consider s o m e n u m e r i c a l e x a m p l e s . I f D ~1- • 1 m o d 9 is c u b e - f r e e a n d if h D is o d d t h e o n l y a priori possible v a l u e o f / , e s s e n t i a l l y d i f f e r e n t f r o m / , ---- 1 is/~ ---- ~. T h u s ~ h a s a t m o s t o n e g e n e r a t o r a n d U a t m o s t o n e g e n e r a t o r of infinite order. Since n o n - t r i v i a l s o l u t i o n s (i. e. s o l u t i o n s of infinite order) do e x i s t f o r D ---- +-2, • • • • • t h e c o r r e s p o n d i n g g r o u p lI h a s p r e c i s e l y o n e i n f i n i t e g e n e r a t o r . T h e o r e m X , u s i n g t a b l e 2, shows t h a t /, ---- ~ is i m p o s s i b l e f o r D ---- + 6 , • • +-42. H e n c e f o r t h e s e D t h e r e a r e n o n o n - t r i v i a l solutions.
T a b l e 2 s h o w s t h a t t h e r e a r e no e v e n v a l u e s of D w i t h e v e n c l a s s - n u m b e r s in t h e r a n g e ID[ < 50 u n d e r c o n s i d e r a t i o n . If, h o w e v e r , a n y e v e n v a l u e s of D e x i s t w i t h e v e n c l a s s - n u m b e r t h e y could be t r e a t e d as w h e n D is odd.
W e c o n s i d e r l a t e r t h e cases D ~ +- 1 m o d 9.
18. 2~//D. W r i t e
D = 4 J , 2 ~ J . W e c o n s i d e r c o n g r u e n c e s t o p o w e r s of t w h e r e
f = [2, 89 t ~ = [2, 6], t 3 = [ 2 ] .
T h e o r e m X I . I f 2211D and (14) is soluble then either tt is a quadratic residue of 4 or (I) i f J ~ - - 1 m o d 4,
/~ := 5 - - 5 , 1 - - 2 5 , 1-}-5 2, 1 + 5 - - 5 2 m o d t 7 , (21) or ( I I ) i f T ~ + 1 m o d 4,
3 2 1 + 5 + 8 9 1 + 5 + ~ 5 2 m o d t7 ( 2 2 )
/~ ~ - - 1 + 8 9 2, 3 + ~ 5 ,
As (x, t) = (y, t) = 1 it is e a s y t o see t h a t o n e of t h e t h r e e cases holds (i) 2~xy, 2/t (ii) 27~xyt (iii) 2/x, 2//y, 2ft.
The Rational Solutions of the Diophantine Equation Y~ ~ Xa--D. 257 (i) 2~xy, 2It. This is a n a l o g o u s to case (i) in t h e previous t w o sections: W e h a v e x :~ x a ~ y2 ~ 1 m o d 8 a n d so/~a2 __ x _ t 2 6 ~ 1 rood 4 i. e./~ is a q u a d r a t i c residue of 4.
(ii) 2r Here x ~ x s = y 2 + D t G ~ l + D ~ - ~ 5 rood 8 a n d so tto~ 2 = x--t26 ~ 5 - - 6 rood 8 .
B u t ~ is p r i m e to t since y is prime, to 2 a n d so
( I / a ) ~_ 1, 1-{--6, 1+ 89 2, 1-{-6-1--89 ~ m o d t a = [ 2 ] . Hence, b y l e m m a 5,
(1/~) 2 ~ 1, 1 + 2 6 + 6 2 , 1+J6=l-62, 5 + 3 J 6 m o d t 7 . (23) T h u s if J ~ g= 1 m o d 4, # is a q u a d r a t i c residue of t 7 a n d afortiori of 4 a n d if J ~ - - 1 m o d 4 one of t h e congruences (21) holds.
(iii) 2/x, 2//y, 27~t. W r i t e x = 2x', y = 2y' so t h a t 2~y' a n d 2x 'a ~-- y'2-+-Jt6 ~ l + J rood 8 . H e n c e J - - I : 3 m o d 8 a n d
4/x if J ~ 7 m o d 8 .
(2~)
x - ~ 2 x ' ~ l + J m o d 8 if J ~ 1 m o d 4 . L e t a -~ 89 ' so t h a t a ' is p r i m e to t a n d
/ ~ ' ~ = (2/6~)2(x--t26) ~ - - J + x 5 2 / 4 m o d t 7 . (25) W e n o w consider t h e t w o cases J ~ 7 rood 8 a n d J ~ 1 rood 4 s e p a r a t e l y .
(iiil) J ~ 7 rood 8. T h e n (24) implies either t h a t t h e r i g h t h a n d side of (25) is c o n g r u e n t t o 1 rood t 7 so t h a t # is a q u a d r a t i c residue of ff a n d a f o r t i o r i of 4 or t h a t it is c o n g r u e n t t o 1 + 6 ~ rood ff a n d t h e n (2I) holds.
(iii2) J ~ 1 rood 4. B y (24) t h e right h a n d side of (25) is c o n g r u e n t to - - J + ( l + J ) 6 2 / 4 rood t 7. Since (1/~') is p r i m e to t, (1/a') 2 satisfies one of t h e con- g r u e n c e s (23) a n d h e n c e (22) holds.
This concludes t h e p r o o f of t h e t h e o r e m .
C o r o l l a r y 1. I f 22//D and/~ is not a quadratic residue of 4 at least one of (14) or (15) is insoluble.
F o r (21) is i n c o m p a t i b l e w i t h t h e set of c o n g r u e n c e s o b t a i n e d f r o m (22) b y writing - - 6 for 6.
I n p a r t i c u l a r , precisely as corollary 2 of t h e o r e m V I I I is d e r i v e d f r o m corollary 1, we h a v e here also
17. Acta mathematiza, 82. I m p r i m 6 le 9 mars 1950.
258 J . W . S . Cassels.
C o r o l l a r y 2. I f 2~//D, D =i~ • 1 rood 9 is cube-free and if h D i8 odd, non-trivial solutions do not exist for both y2 = xa =~ Dt 6. I f non-trivial solutions do exist for one of these equations then the corresponding group U has precisely one generator of infinite order.
W e n o w consider s o m e n u m e r i c a l e x a m p l e s .
N o n - t r i v i a l solutions a c t u a l l y e x i s t f o r t h e v a l u e s D = + 4 , - - 1 2 , + 2 0 a n d - - 3 6 w h i c h s a t i s f y t h e c o n d i t i o n s of c o r o l l a r y 2. H e n c e t h e r e is p r e c i s e l y one infinite g e n e r a t o r for e a c h of t h e c o r r e s p o n d i n g g r o u p s 1I. Moreover, b y t h e s a m e corollary, t h e r e are no n o n - t r i v i a l solutions for D - ~ - - 4 , + 1 2 , - - 2 0 , + 3 6 .
W e l e a v e for l a t e r c o n s i d e r a t i o n t h e cases D ~ ~_ 1 rood 9.
19. 2a//D. As (x, t) = (y, t) = 1 it is n o t difficult to see t h a t one of t h e t h r e e cases holds: (i) 2 t x y , 2/t (ii) 2 f x y t (iii) 2~Ix, 2~/y, 2tt. Of t h e s e (iii) gives rise to t h e n e w p o s s i b i l i t y t h a t [x--t~6] is n o t t h e s q u a r e of a n ideal.
Since we a r e e x c l u d i n g p e r f e c t cubes a t p r e s e n t , t h e o n l y v a l u e s in t h e r a n g e IDl ~ 50 u n d e r c o n s i d e r a t i o n a r e D = ~:24, 2=40. F o r all t h e s e D = + 2 a D * w h e r e D * is o d d a n d c u b e - f r e e a n d hD. ~-= 1. As b e f o r e let A a = D * . W e consider c o n g r u e n c e s t o p o w e r s of t a n d u, t h e f i r s t a n d s e c o n d d e g r e e p r i m e d i v i s o r s r e s p e c t i v e l y of 2, w h e r e
t = [2, I + A ] , t 2 -= [4, D * - - A ] , t ~ = [S, D * - - A ] ,
tt = [2, I + A + A 2 ] , n z --- [4, I + D * A + A 2 ] , u ~ = [8, I + D * A + A Z ] . B y t h e discussion in w 12 t h e o n l y a priori possible v a l u e s of/~ a r e / ~ = 1, ~ in eases (i) a n d (ii) a n d / ~ = 2, 2 9 in case (iii) w h e r e 2 > 0 a n d [2] = u. T h e g r o u p U h a s no, h a s one or h a s t w o g e n e r a t o r s of i n f i n i t e o r d e r a c c o r d i n g as no, o n e or all t h r e e v a l u e o f / ~ e s s e n t i a l l y d i s t i n c t f r o m 1 c a n a c t u a l l y o c c u r in (14). Case (i) is c o m p l e t e l y a n a l o g o u s to ease (i) p r e v i o u s l y . I t implies t h a t / ~ is a q u a d r a t i c r e s i d u e of 4 i. e. t h a t # = 1 a n d so it m a y b e n e g l e c t e d in t h e r e s t of t h i s w
W e n o w t r e a t D = + 2 4 , D = - - 2 4 , D = ~=40 s e p a r a t e l y . W e shall s h o w t h a t t h e c o r r e s p o n d i n g g r o u p 1I h a s no, t w o a n d one g e n e r a t o r of infinite o r d e r r e s p e c t i v e l y . D ~ - - + 2 4 , A 3 = D * = 3. (ii) 27~xyt. H e r e x ~ x 3 ~ y ~ - ~ 1 rood 8 a n d so
i ~ 2 -= x - - 2t2A
1 - - 2 A m o d 8 3 mo~t t 3 .
B u t ~ is p r i m e to t, since y is p r i m e to 2, a n d so ~2 ~ 1 rood t a ( l e m m a 5). H e n c e
T h e R a t i o n a l S o l u t i o n s o f t h e D i o p h a n t i n e E q u a t i o n y 2 ~ X a _ D . 2 5 9
# ~ 3 m o d ia. This is clearly impossible for # ~ 1 a n d it is also false for t h e o n l y o t h e r possibility # ~ ~ -= A 2 - - 2 ~ - - 1 m o d t a.
(iii) 2//x, 22/y, 2 f t . L e t x -~ 2x' where x'.is o d d a n d l e t ~ l d e n o t e either 1 or~. T h e o n l y a priori possibility for # is ~, 2 where ~ > 0 a n d [4] -~ u, s a y ), ~ 2(A - - 1) -1. T h e n
~l~s -- x ' - - t s z l ~ x ' - - A m o d 4 . A - - 1
N o w ~ is p r i m e to u since x ' - - t S A is b y l e m m a 2 a n d so ( l / s ) ~ 1, I + A , A rood u b y l e m m a 4. H e n c e ( I / a ) s _ ~ l, - - A , - - I + A m o d u s' b y l e m m a 5, a n d t h e n
~1 ~ ( x ' - - A ) ( A - - I ) ( 1 / ~ ) 2 m o d u s 1 - - x ' + x ' A , - - l + ( 1 - - x ' ) A , x ' - - A m o d u s .
Since x' is o d d this c o n g r u e n c e is impossible b o t h for ~1 ~ I a n d for V~ ~ ~ ~ A s _ 2 I + A m o d u s.
H e n c e w h e n D ~ + 2 4 n o n e of t h e a p r i o r i possible values of # essentially different f r o m 1 a c t u a l l y c a n occur, i . e . 1I has no g e n e r a t o r s of infinite order.
D - ~ - - 2 4 , A a D * = - 3 . (if) 2~xyt. ' S o l u t i o n s of this t y p e do exist w i t h
# = 7- A n e x a m p l e is
l a - - 5 2 --- - - 2 4 since 1 + 2 A ~ / ~ s is n o t a p e r f e c t s q u a r e L
(iii) 2//x, 2S/y, 2 f t . Solutions do exist of this t y p e i. e. w i t h / t : 2 ~ / ( A - - 1 ) w h e r e ~ ~ 1 or y. A n e x a m p l e is
( - - 2 ) a - - 4 s -~ - - 2 4 .
H e n c e for D ~ - - 2 4 solutions exist for a t least two, a n d so for all t h r e e a priori possible values of # essentially d i s t i n c t f r o m 1 i. e. 1I hs t w o g e n e r a t o r s of infinite order.
D ~ i 4 0 , A ~ ~- D * =- 5. (if) 2 $ x y t . H e r e x ~ x a ~ yS ~ 1 m o d 8. T h e o n l y a p r i o r i possible v a l u e of # essentially different f r o m 1 (which we neglect) is # ~ ~.
I f this a c t u a l l y o c c u r r e d we should h a v e
~7~ ~ =- x ~ 2tSA ~ l + 2 A m o d 4 , a n d so
~ l + 2 A , 2 - - A , 1 - - A m o d u s
b y a f a m i l i a r a r g u m e n t . This is a c o n t r a d i c t i o n since ~ : = 1 - - d A + 2 z l ~ =~ - - 1 - - A m o d u s.
I 1 -}- 2A ~ -- 1 m o d t a w h e r e a s a p e r f e c t s q u a r e is c o n g r u b n t to + 1 b y l e m m a 6.
260 J . W . S . Cassels.
(iii) 2~Ix, 22/y, 2 / t . Solutions do exist of this t y p e , i . e . with /~ : ~12, for b o t h D : § (see t a b l e 1). B y t h e g r o u p p r o p e r t y o f / t a n d since/~ = ~ was s h o w n n o t to occur, solutions with/~ = 2 a n d tt : ~]2 c a n n o t b o t h o c c u r for t h e s a m e value of D.
H e n c e w h e n D = -+ 40 precisely one of t h e a priori possible values of/~ essentially different f r o m 1 a c t u a l l y occurs i . e . lI has one g e n e r a t o r of infinite order.
20. 24//D. Since (x, t) : (y, t) : 1 it is n o t difficult to see t h a t one of t h e t h r e e cases holds (i) 2 f x y , 2It (ii) 27~xyt (iii) 22/x, 22//y, 27~t.
T h e o n l y values of D to discuss are D : -+ 16, -+48. Since for these D zl~ -+ 1 rood 9, G has no f a c t o r s which are n o t a l r e a d y f a c t o r s of D * : E F 2 a n d hD, = l, t h e discussion of w 12 shows t h a t t h e o n l y a priori possible v a l u e o f / t essentially dissimilar to 1 is/~ : ~. T h e r e is t h u s one or no g e n e r a t o r of infinite order a c c o r d i n g as solutions of infinite order (i. e. n o n - t r i v i a l solutions) do or do n o t occur.
D - - -+48. N o n - t r i v i a l solutions o c c u r for b o t h these values of D (table 1).
H e n c e for b o t h D : -+ 48 t h e g r o u p 1I has one infinite g e n e r a t o r .
D - - + 1 6 . A n : D* = 2. W e shall show t h a t no n o n - t r i v i a l solutions exist.
I t is e n o u g h to show t h a t n o n e exist for w h i c h / t = ~. W e consider c o n g r u e n c e s t o p o w e r s of f where
f = [2, A], t ~ = [2, AS], t ~ = [ 2 ] .
As before, this implies t h a t / ~ is a q u a d r a t i c residue of 4, i. e.
(i) 2 f x y , 2It.
t h a t / ~ - - 1.
(ii) 2,r H e r e x ~ : x 3 ~ y 2 ~ 1 m o d 8 a n d so i f # = ~ we should h a v e
~o~ ~ = x • 2 :~ 1-+2A m o d 8 . H e n c e b y a familiar a r g u m e n t
~] ~ l + 2 A , 5 + A "~, l + 2 A - 4 - 3 A 2, 1-4-2A 2 m o d t 7 . This is a c o n t r a d i c t i o n since ~ = - - I + A .
(iii) 22/x, 22//y, 2~t. W r i t e x = 22x ', y = 2~y ' so y ' is odd. T h e n 4x'3 : y ' 2 - + t n .
T h e u p p e r sign is impossible rood 8 a n d if t h e lower is to hold we h a v e 2/x'. F u r t h e r , if /~ - - 0 we h a v e (taking o n l y t h e lower sign)
~o~ 2 = x + 2t2A = 4x' + 2t2A : (A2)2(t2+x'A2.) .
I f 4Ix', t 2 + x ' A 2 ~ 1 m o d 4 a n d so ~ w o u l d be a q u a d r a t i c residue of 4, w h i c h is
The Rational Solutions of the Diophantine Equation y2 = X a _ D . 261 u n t r u e . I f 2//x' we also h a v e a c o n t r a d i c t i o n since t h e n t e ~ x ' A 2 ~ 1 + 2 A e m o d t ~ = 4[2, A] a n d so
~ l + 2 A 2, l ~ 2 A ~ 3 A ~ , 5 ~ A 2, l + 2 A m o d i f , w h e r e a s r 1 ~ - - I ~ A .
T h i s concludes t h e p r o o f t h a t t h e r e a r e no n o n - t r i v i a l solutions w h e n D ~ ~ 1'6.
21. 25//D. Since (x, t) ~ (y, t ) = 1 it is e a s y t o see t h a t e i t h e r (i) 2~xy, 2It or (ii) 27~xyt.
T h e o n l y v a l u e s of D to discuss arc D ~ =E32. H e r e A 3 ~ D* ~ 4, hD, ~ 1.
T h e o n l y a priori possible v a l u e of # o t h e r t h a n 1 is # ~ ~ / a n d so 1I h a s one or no g e n e r a t o r s of infinite order a c c o r d i n g as solutions of infinite o r d e r (i. e. n o n - t r i v i a l solutions) do or do n o t exist, W e shall show t h a t no n o n - t r i v i a l solutions exist. I t is e n o u g h to show t h a t n o n e e x i s t w i t h /z -~ ~.
W e consider c o n g r u e n c e s to p o w e r s of t w h e r e t = [2, ~A~], t 3 = [2, ~ ] , t ~ = [ 2 ] .
(i) 2~xy, 2It. As before this implies t h a t # is a q u a d r a t i c residue of 4, i. e./~ ~ 1.
(ii) 2r H e r e x ~ x ~ y 2 _ ~ 1 m o d 8 a n d so w h e n # = ~ we h a v e
~ z _~ x:j:2t2A ~_ 14-2A rood 8 1 - - 2 A m o d t 7 . H e n c e
~ 1 2A, I + A 2, 5 + A , 1 A - - A 2 m o d U , b y a k n o w n a r g u m e n t . T h i s c o n t r a d i c t s V ~ l ~ - 8 9
22. D ~ =E 1 rood 9, D cube-free, h D odd. All t h e r e m a i n i n g v a l u e s of D in t h e r a n g e D 1 __< 50 w h i c h are n o t p e r f e c t cubes fall in t h i s c a t e g o r y . W e shall consider c o n g r u e n c e s t o p o w e r s of ~ a n d ~ where
~ 2 ~ _ [ 3 ] , ~ - - [3, D 6 ] .
B y t h e a r g u m e n t of w 12 t h e o n l y a priori possible v a l u e s of # are
# - - 1, V, ~t, ~ , w h e r e Z is d e f i n e d b y
T h e v a l u e s 2, 2, t o c c u r if a n d o n l y if 3/y. T h e g r o u p U h a s no, or one, or t w o g e n e r a t o r s of infinite o r d e r a c c o r d i n g as no, or one, or all t h r e e of t h e v a l u e s of # o t h e r t h a n 1 a c t u a l l y occur.
262 J . W . S . Cassels.
S u p p o s e 3/y so t h a t # = ~ , ) t w h e r e ~,--- 1 or ~. T h e n 3 f t , a n d so x ~ x 3 =-- D t 6 ~ D rood 3. H e n c e
~ , ~ = x - - t 2 6 ~ D - - J z!_: 0 m o d ~25, 0 rood ~$.
Since r - ~ a n d c o n s e q u e n t l y ~2 ~ 1 m o d ~, we d e d u c e
~, ~ ( D - - 6 ) ~ - ' m o d ~. (26)
W e n o w p r o v e t w o g e n e r a l t h e o r e m s a c c o r d i n g as ~] ~ ~ 1 m o d v.
T h e o r e m X I I . T h e group 11 has at most one infinite generator i f (I) D ~ ~: 1 m o d 9, ( I I ) D is cube-free ( I I I ) h D is odd (IV)~ ~ - - 1 rood r.
F o r if ~ ~ - - 1 m o d r t h e c o n g r u e n c e (26) c a n n o t be t r u e b o t h w i t h ~ -= 1 a n d w i t h ~ = ~l i. e. n o t all t h r e e v a l u e s o f / ~ o t h e r t h a n 1 a c t u a l l y occur.
T h e o r e m X I I I . A t least one of the two equations y2 _ xS ~ D t ~ has no solutions with 3/y i f (I) D ~ ~: 1 m o d 9 ( I I ) D is cube-free ( I I I ) h D is odd (IV) ~7 ~ + 1 m o d ~.
F o r if (26) is t r u e either w i t h ~, -=-- 1 or w i t h ~, = z] t h e c o r r e s p o n d i n g c o n g r u e n c e in which t h e signs of D a n d ~ a r e s i m u l t a n e o u s l y c h a n g e d is false both f o r ~ = 1 and f o r ~1 = ~.
W e n o w consider s o m e n u m e r i c a l e x a m p l e s .
T h e conditions of t h e o r e m X I I a r e satisfied f o r t h e following v a l u e s of D, D =- ~ 1 9 , J:28, ~ 3 5 , ~ 4 4 .
Since n o n - t r i v i a l solutions e x i s t for t h e s e D (table l) t h e c o r r e s p o n d i n g g r o u p U h a s p r e c i s e l y one infinite g e n e r a t o r .
T h e r e a r e solutions w h e n D--- - - 1 7 b o t h f o r /~ = ~ a n d f o r /x-= ~ 2 since ( - - 1) 3 _ 42 _ ( _ 2) 3 _ 3 ~ ~ - - 17. H e n c e t h e g r o u p 1I_17 h a s t w o infinite g e n e r a t o r s . F o r D ---- + 17 t h e r e a r e c o n s e q u e n t l y no solutions e i t h e r w i t h / ~ --- ~ ( t h e o r e m V I I I c o r o l l a r y 1) or w i t h # ---- Vl~ ( t h e o r e m X I I I ) . H e n c e t h e r e are no n o n - t r i v i a l s o l u t i o n s a t all for D---- +4-17. S i m i l a r l y for D---- J=37.
W h e n D ~ ~=10 t h e r e a r e no solutions w i t h / ~ ---- ~ b y t h e o r e m X. As a n o n - t r i v i a l solution ( - - 1 ) 3 32 _ _ 10 does e x i s t for D ---- - - l 0 t h e c o r r e s p o n d i n g g r o u p 1110 h a s p r e c i s e l y one infinite g e n e r a t o r . B y t h e o r e m X I I I t h e r e a r e c o n s e q u e n t l y no solutions w i t h 3/y f o r ~D ---- ~ l0 i. e. n o n e w i t h # ---- ~1~. Since t h e r e a r e also n o n e w i t h / ~ ~- ~ t h e r e a r e n o n e a t all.
]By a p r e c i s e l y s i m i l a r a r g u m e n t t h e g r o u p 1146 h a s no a n d t h e g r o u p
114~
p r e c i s e l y one infinite g e n e r a t o r . W e n o t e f u r t h e r t h a t t h e s o l u t i o n f o r D -~ - - 4 6 ,
T h e R a t i o n a l S o l u t i o n s of t h e D i o p h a n t i n e E q u a t i o n y 2 ~ X a _ D . 263 ( - - 7 ) 8 - - 5 1 2 ~_ --46.26
m u s t correspond to a # other t h a n 1 since 3/y, a n d hence to t h e only other occurring value of ,u. Consequently t h i s value of ,u is also a q u a d r a t i c residue of 4, i.e. # is a q u a d r a t i c residue of 4 for all solutions w i t h D - ~ --46. This implies t h a t 2It a n d hence X -= x/t 2, Y ~ y / t 3 c a n n o t be integers (ef. w 15).
F i n a l l y D -- -}-26 has a group 1I w i t h two generators. H e n c e D ~ --26 can h a v e a group 1I w i t h at most one generator, b y t h e o r e m X I I I a n d so precisely one, as non-trivial solutions do exist.
23. D = G 8. W e discuss now t h e case w h e n D is a perfect cube. B y t h e r e m a r k s at the beginning of p a r t I I we m a y assume
G s q u a r e - f r e e , 3/~G.
The roots ~1, 82, ~a of ~8--D ~ 0 are G, 5G, 52G where 5 is a complex cube root of u n i t y . Since ~ , (~2, 53 are n o t all conjugate we m u s t use t h e " t r i p l e t s " i n t r o d u c e d in w 5 to define the group (~. W e recollect t h a t t h e set of triplets ~ (x--t2(~i} f o r m t h e multiplicative group 6J w h e n squared f a c t o r s are ignored. W e shall first prove
L e m m a 9. The set of triplets {x--t2~j} corresponding to solutions with 3~y f o r m a subgroup G ~ of ~ with one less generator.
We n o t e t h a t R(~I) = R(1), R(52) = R(Sa) = R(Q) a n d t h a t (l--Q) is a prime divisor of 3 in R(5 ) satisfying
( 1 - 5 ) 2 = - 3 5 .
W r i t e Oj ~- x--t2(~j so t h a t
01 -~ x--Gt2, ' 02 = 93 -~ x - - s G t 2 ,
where t h e bar denotes the complex conjugate. Now it would follow from 3./02 t h a t 3Ix, 3/Gt 2 c o n t r a r y to (x, t) ~ 1 a n d 3~G. H e n c e (1--5)2202 a n d it is easy to verify t h a t (1-5)202 or (~-5)//02 according as a/y or
a/y.
Clearly t h e set of {Oj} for which ( 1 - - 5 ) f 0 ~ form a subgroup (~o of (~ which either coincides w i t h (~ or is of i n d e x 2, i. e. t h e set of {Oj} corresponding to solutions w i t h 32y does this. F u r t h e r (~0 c a n n o t coincide with ~ since there is always the trivial solution (x, y, t) =- (G, O, 1) which does n o t belong to G ~ Since all t h e elements of (~ are of order 2 it follows t h a t (~0 has one fewer generator t h e n (~, which proves t h e lemma.1 el. w 8.
264 J . w . s . Cassels.
Suppose now t h a t 3/~y so t h a t all the Oj are p r i m e to 3 (and non-zero). W e shall find a b o u n d e d set of triplets {/tj} ---- {/~1,/~2, ~2) such t h a t
0 ~ = / ~ a 2, a E R ( 1 ) ; 02----/~2x -+, ~ E R ( ~ )
always holds for one of t h e set, a n d i n d e e d with /q,/~2, a, ~ all integers. Since
0 1 ~ 1 - ~ 0 2 - ~ 0 2 0 3 : 0
t h e c o m m o n f a c t o r of an.y two of t h e Oj m u s t also divide t h e t h i r d . This c o m m o n f a c t o r m u s t also divide 02--01 : (1--o)Gt 2 a n d 02--~01 : (1--Q)x a n d so since (x, t) = 1 a n d t h e Oj are p r i m e to 3 : - --~2(1--~))2 we h a v e
(01, 02, 03) = (x, G) = K > 0 ( s a y ) .
W r i t e 0j : K ~ j where t h e ~vj are co-prime in pairs. N o w y2 :_ 010203 : K3~vl~v3 a n d K , being a divisor of G, is square-free. H e n c e y : K2b, where b E R(1) is an i n t e g e r a n d
~ 1 ~ 2 ~ 2 = ~/71~92q73 = K b 2 "
Since K > 0 a n d t h e ~ j a r e co-prime in pairs it follows t h a t t h e r e are integers H > 0, H E R(1) a n d v E R(ff) such t h a t
q~l -~ Ha2, a E R(1); ~v 2 ~- ~ - - v~ 2, ~ E R(C); K ~- Hv~ - H N ( s a y ) , i . e .
01 : K H a 2, 02 --- Oa = K v ~ 2 . W e m a y t h e r e f o r e p u t
{/~j} ---- {,ul, ,u2,/~a} : { K H , K v , K ~ } . (27) F u r t h e r , b y eliminating x b e t w e e n 01 a n d 02 we o b t a i n
i. e .
w h e r e
K v a ' 2 - - K H a 2 = ( 1 - - e ) G t 2 H a e - j - ( 1 - - ~ ) J t 2 ~- vo~ 2, t :4: O, a # 0 , G = J K ~- H J N , N ~- r~ , H > O.
(28) (29) Conversely a solution of (28) gives a solution of t h e original e q u a t i o n y2 _~ x ~ _ G S t 8 w i t h 3/~y. Since t h e groups @0 or (~ i n v o l v e triplets t h e y are difficult to handle.
W e n o w show t h a t we m a y use a g r o u p n o t involving triplets.
T h e o r e m X I V . T h e values o f v / H f o r which (28) is soluble f o r m a m u l t i p l i e a t i v e g r o u p ~ i f squared factors are ignored w i t h the s a m e n u m b e r o f generators as there are i n d e p e n d e n t generators of i n f i n i t e order i n 1I.
The Rational Solutions of the Dioi)hantine Equation y2 = X a _ D . 265 Since t h e set of t r i p l e t s
{~gl' ~2' /~3}
of t h e f o r m which a c t u a l l y o c c u r w i t h s o l u t i o n s of y2 = x a G 3 t 6 f o r m t h e m u l t i p l i c a t i v e g r o u p 650 w h e n s q u a r e d f a c t o r s a r e i g n o r e d t h e v a l u e s of l~2//~1 ~ v / H do f o r m a m u l t i p l i c a t i v e g r o u p ~ w h e n s q u a r e d f a c t o r s a r e ignored. N o w (v, H ) = 1 since v~H - - N H is a divisor of G, w h i c h is s q u a r e free.T h u s , since H > 0, t h e r a t i o v / H d e t e r m i n e s v a n d H u n i q u e l y i. e. b y (27) a n d (29) it d e t e r m i n e s {#j} u n i q u e l y . H e n c e ~ is i s o m o r p h i c to 650 a n d , in p a r t i c u l a r , t h e t w o g r o u p s h a v e t h e s a m e n u m b e r of g e n e r a t o r s , i. e. one f e w e r t h a n 65 b y l e m m a 9.
T h e t h e o r e m n o w is a n i m m e d i a t e c o n s e q u e n c e of t h e o r e m I I I since 6 3 - - D - - 0 h a s j u s t one r a t i o n a l r o o t .
C o r o l l a r y 1. I f (28) is insoluble except, possibly, when v -~ H ~ 1, the group li has no generators of infinite order.
F o r t h e n ~ h a s no g e n e r a t o r s . I n p a r t i c u l a r , b y t h e o r e m V,
C o r o l l a r y 2. I f D 4= - - 1 and (28) is insoluble except, possibly, when v ~ H ~- 1 there are no non-trivial solutions of y2 ~ x3 G3t%
W e n o w p r o v e a useful l e m m a .
L e m m a 10. I f v -= ~ 1 and (28) has solutions then v H : N o r m Z where Z E R O / 3 ).
F o r s u p p o s e c~ ~-- e + f ~ w h e r e e a n d f are r a t i o n a l integers. B y e q u a t i n g coef- ficients of 1 a n d ~ in (28) a n d e l i m i n a t i n g t we o b t a i n
~,Ha 2 .~ e2 + 2ef--f~ = ( e - F f ) ~ - - 2 f f w h i c h p r o v e s t h e l e m m a .
Since we are a s s u m i n g 37~G t h e o n l y v a l u e s of D ~- G 3 in t h e r a n g e [D I < 50 are D --~ + 1 a n d D = :~ 8. T h e y are b o t h c o v e r e d b y t h e g e n e r a l t h e o r e m d u e t o N a g e l l [29].
T h e o r e m X V . I f every p r i m e divisor p > 0 of G is of the f o r m 12n~-5, the equations y2=_ x a • 6 have no solutions of infinite order (and so no non-trivial solutions except when D = - - 1 ) .
F o r N o r m v -~ v~ = N ~ 1 is i m p o s s i b l e in R(~) if t h e f a c t o r s of N are to b e of t h e r e q u i r e d t y p e . H e n c e v = + 1 , ~ , -~-e 2 a n d b y a b s o r b i n g f a c t o r s ~ = (~2)2 or ~2 in ~2 we m a y a s s u m e t h a t ~ = + 1 . H e n c e , b y l e m m a 10, v H - ~ N o r m ~, Z E R(~/3) a n d t h i s a g a i n implies v H = 1, i. e. v = H = 1 (since H > 0). T h e o r e m X V n o w follows f r o m c o r o l l a r y 1 t o t h e o r e m X I V .
