THE RATIONAL SOLUTION OF THE DIOPHANTINE EQUATION
y 2 X S D . By J. W. S. CASSELS
of MANCHESTER.
A d d e n d a a n d C o r r i g e n d a .
1. I o v e r l o o k e d t h a t t h e o r e m I V was first p u b l i s h e d b y Nagell (my reference [32]).
2. Dr. E. S. Selmer tells me t h a t m y c o n j e c t u r e I of w 25 c a n n o t be correct, e. g. for D : --612, y2 ~ x~+612ts. H e r e a possible # is t h e u n i t ~ : 1 - - 1 6 ~ - 4 ( ~ 2 where 8 a ~ 61 a n d t h e c o n g r u e n c e
X_~.t2~2 ~ ~]~2
is soluble m o d u l o a n y power of 2 (the only r e l e v a n t modulus), i n d e e d with x = l, t --= 2. On t h e o t h e r h a n d Dr. Selmer has p r o v e d t h a t y2 = xa~_612ts is insoluble.
3. An ingenious m e t h o d of i n v e s t i g a t i n g t h e g e n e r a t o r s of y z = X 3 _ D using o n l y t h e p r o p e r t i e s of q u a d r a t i c fields has j u s t been given b y V. D. P o d s y p a n i n (Mat. Sbornik 24 (1949) 391-403). H e gives a t a b l e of g e n e r a t o r s for ID] < 90 b u t a . c o m p a r i s o n w i t h m y t a b l e for IDI _< 50 shows a n u m b e r of e r r o r s : - -
(i) F o r D = 48 P o d s y p a n i n gives t w o g e n e r a t o r s (4, 4, 1), (73, 595, 3) b u t actually t h e p a r a m e t e r of t h e second solution is j u s t 3 t i m e s t h a t of t h e first.
(ii) No g e n e r a t o r s are given b y P o d s y p a n i n for D = • 50 a n d insufficient g e n e r a t o r s for D ~ - - 1 5 , - ~ 3 9 .
4. T h e following two misprints h a v e o c c u r r e d : - -
(i) P a g e 265, proof of l e m m a 10: F o r " v H a 2 - ~ e e + 2 e f - - f 2 - ~ ( e + f ) z - - 2 f 2
" r e a d " v H a 2 ~- e 2 + 2 e f - - 2 f ~ ~-- (e-~f)2--3f ~''.
(ii) P a g e 271, line 6: F o r " u n i t s for all in W o l f e " r e a d " u n i t s for all D ___ 100 in W o l f e . "
