A(k)= (x, zl ... zk-1, x~, z~+l ... xn)

A R K I V F O R M A T E M A T I K B a n d 3 n r 17

G o m m m l i c a t e d 1 D e c e m b e r 1954 b y T. N A O E L L

O n a c l a s s o f D i o p h a n f i n e e q u a t i o n s

B y LARS

FJELLSTEDT

HURWITZ [1] 1 h a s s t u d i e d t h e D i o p h a n t i n e e q u a t i o n

x~ + ~ + ... + ~ = x . =l x~ ... ~ .

(1)

F o r n = 2 m o r e g e n e r a l e q u a t i o n s of t h i s t y p e h a v e b e e n i n v e s t i g a t e d b y BAR-

~ . s [2] a n d MILLS [3, 4]. This p a p e r deals w i t h t h e e q u a t i o n

(X~ "~ a.:) 9" -~ 6 = x . x I x 2 - . . •n, (2)

i=1

w h e r e t h e n u m b e r s a~ a n d c a r e r a t i o n a l i n t e g e r s , c > 0. W e s h o w t h a t w i t h s o m e a l t e r a t i o n s t h e m e t h o d u s e d b y Hul~WlTZ o n e q u a t i o n (1) also a p p l i e s t o e q u a t i o n (2). T h e r e s u l t s o b t a i n e d a r e a n a l o g o u s t o t h o s e of Hln~WiTZ.

W e n o t e t h a t if a n y of t h e p o l y n o m i a l s (xi + at) 2 + c = 0

is r e d u c i b l e , e q u a t i o n (2) is s o l v a b l e for e v e r y v a l u e of x. S i m i l a r l y if t h e r e e x i s t s a s o l u t i o n i n w h i c h one a t l e a s t of t h e n u m b e r s xl, x 2 .. . . . x~ is zero.

I n t h e following lines we consider o n l y s o l u t i o n s i n w h i c h t h e n u m b e r s x 1, x~, . . . , x~ a r e a l l 4: 0.

S t a r t i n g f r o m a s o l u t i o n

A = (x, x 1, x~ . . . xn)

we c a n f i n d a n e w s o l u t i o n A (k) in t h e following w a y . I n (2) we s o l v e for xk.

Since t h e e q u a t i o n c o n s i d e r e d is of t h e s e c o n d d e g r e e a n d since i t is s a t i s f i e d b y xk, t h e o t h e r r o o t is

I

X k ~ X " X 1 "** X k - 1 X k + l " "

xn--2

a k - - X k . (3)

I t is o b v i o u s t h a t x~ is a n integer. T h u s

A(k)= (x, zl ... zk-1, x~, z~+l ... xn)

is a s o l u t i o n of (2). T h e s o l u t i o n s A (k), k = 1, 2 . . . n a r e s a i d t o b e associated w i t h t h e s o l u t i o n A .

1 Numbers in brackets refer to the bibliography at the end of this paper.

L. F J E L L S T E D T , O n a class

of Diophantine equations

A s e q u e n c e of s o l u t i o n s A~, A~ . . . of (2) w i t h t h e p r o p e r t y t h a t e v e r y solu- t i o n is a s s o c i a t e d w i t h t h e p r e c e d i n g one f o r m s a

chain o] solutions.

I t is e v i d e n t t h a t x h a s t h e s a m e v a l u e for e v e r y s o l u t i o n i n a c h a i n .

T h e n u m b e r

~ [ x i l i = 1 , 2 . . . n is c a l l e d t h e

weight

of t h e s o l u t i o n (x, xj . . . xn).

A s o l u t i o n of w e i g h t h is s a i d t o b e

/undamental

if t h e w e i g h t of all t h e s o l u t i o n s a s s o c i a t e d w i t h i t is _-> h.

A f u n d a m e n t a l s o l u t i o n is s a i d t o

generate

t h e c h a i n t o which i t belongs.

W e h a v e t h e following r e s u l t : T h e o r e m I .

Every solution o/ (2)

(X, Xl, X2, . . . , Xn)

with

Xl, x 2 . . .

x n all ~-0, belongs to a chain generated by a [undamental solution.

P r o o f . S u p p o s e t h a t a s o l u t i o n A of (2) is n o t f u n d a m e n t a l . T h e n i t h a s a n a s s o c i a t e d s o l u t i o n whose w e i g h t is less t h a n t h e w e i g h t of A . I f t h e n e w s o l u t i o n is n o t f u n d a m e n t a l w e c a n r e p e a t t h e process. Since t h e w e i g h t is a n a t u r a l n u m b e r w e m u s t f i n d a f u n d a m e n t a l s o l u t i o n a f t e r a f i n i t e n u m b e r of s t e p s .

To solve (2) c o m p l e t e l y we h a v e o n l y t o d e t e r m i n e t h e f u n d a m e n t a l solutions.

O u r m a i n r e s u l t will b e t h a t t h e n u m b e r of d i f f e r e n t f u n d a m e n t a l s o l u t i o n s is f i n i t e a n d f u r t h e r m o r e we will g i v e i n e q u a l i t i e s for t h e s e solutions.

I f (x, x I . . . xn) is a s o l u t i o n of (2) in w h i c h t h e n u m b e r s

x~,, x~

. . .

x~r

a r e n e g a t i v e , (x, I x l l , ]x~l . . . I x . I ) is o b v i o u s l y a s o l u t i o n of t h e e q u a t i o n d e r i v e d f r o m (2) b y c h a n g i n g signs for % , a, . . . a, r. F u r t h e r m o r e i t is clear t h a t if w e can p r o v e for t h i s n e w e q u a t i o n t h a t t h e n u m b e r of d i f f e r e n t f u n d a m e n t a l solu- t i o n s is f i n i t e , t h e s a m e r e s u l t h o l d s for (2). H e n c e we s u p p o s e i n t h e s e q u e l t h a t all t h e n u m b e r s

xl, x ~ ... x~

a r e > 0 .

I f t h e w e i g h t of t h e s o l u t i o n A is n o t g r e a t e r t h a n t h a t of A (k), we g e t f r o m (3) on m u l t i p l i c a t i o n w i t h xk

x~ x~ + x~ + 2 ak xk = x ~ x~.

i l l

F r o m (2) we see t h a t xk a n d x~ h a v e t h e s a m e sign a n d t h e r e f o r e

2x~ + 2akxk <=X f i X,

(4)

iffil

for k = 1, 2 . . . n.

L e t us c o n s i d e r t h e e x p r e s s i o n

(X" X 1 X 2 ' ' " Xk_ 1 X k + l ' ' " X n -- 2 (X k ÷ a k ) ) 2 = X 2 X 2 " ' " Z2k_l X2k+l "'" X 2 *~

+ 4 [ x ' x l x 2 " " x ' ( i+a~t-(x~+ak)2]xe/

ARKIV FOR M A T E M A T I K , B d 3 n r 17 w h i c h c a n b e w r i t t e n

x" x 1 x ~ " " xn - 2 (x~ + a~ xk) = x l x 2 " " xn ~/x ~ : -

4tk

w h e r e

(5)

(1-t-ak~[~(x~÷a')2÷c] ~=1

, x~. (6)

~k ~ 2 2 . 2 ; 2

X l X2 •.

i = ] , 2 . . . n. {7)

F o r s o m e i n d e x ~ w e h a v e

o r

w h e r e

F r o m (5) we g e t o n a d d i t i o n

Tbg~ '=1 ~ X,--~ '=1

~ (x~ + a,x,)= ,=1 ~-[ x,. (iffil ~ ] / ~ ) '

(n - 2) x x~ + 2 T = x,.

t = 1 ' = 1 ' =

T= ~ (x~a~+a~)÷c.

(s)

W e d i s t i n g u i s h t w o cases a c c o r d i n g as T is g 0 o r > 0.

S u p p o s e first t h a t T =< 0. T h e n we g e t f r o m (8)

(n-2)x>= ~ ~/x-~-4t,.

i = 1

A s is e a s i l y seen, i t is sufficient t o c o n s i d e r t h e case tg > 0. I n f a c t , if t < 0 we m u s t h a v e t, < 0 for all i. I t t h e n follows f r o m t h e l e f t - h a n d side of (5) t h a t

xk (ak + xk) < 0, /c = 1, 2 . . . n.

T h i s , h o w e v e r , m e a n s t h a t xk < l akl for a l l k.

I f t , is = 0 , we m u s t h a v e t~:<t, for all i. T h u s we g e t as b e f o r e

xk<:[ak[.

O n s q u a r i n g a n d i n t r o d u c i n g t , i n s t e a d of t, we g e t n2~>_- ( n - 1 ) x 2.

S u b s t i t u t i n g in t h i s i n e q u a l i t y t h e e x p r e s s i o n for t , g i v e n b y (6) we f i n d

[( )(

^a~ ~ (x, + a,) 2 + c - (2;. + a.) 2

)]

~ ( ~ - 1) - - H

x~.

n' 1 + ~ ,-1 - x~,=l ⁽⁹⁾

W e n o w d i s t i n g u i s h t w o cases a c c o r d i n g a s a , is > 0 o r ~ 0. I f a~ ~ 0 we g e t f r o m (9)

L. FJELLSTEDT,

On a class of Diophantine equations

Suppose n o w t h a t

(xm + am)2 ~ (x, + a~) ~, i ~: lz.

T h e n we h a v e

n * [ ( n - 1) (aS + 1) + c] > ( n - 1)

If on t h e o t h e r h a n d a t > 0 we h a v e

H e r e either

2 2 Xp X m

n ] X2 n

n 2 ( n - - l )

(am+ 1)zm + c

e s

+ a , x ~ x ' > (n-- 1) ~ YI z~.

Xp = p t -

'ffil 1

(10)

H o w e v e r , in b o t h cases we h a v e t h e i n e q u a l i t y

X n 2

YIx~. ~ [(n-1)(aL+l)+la, l+c]. (11)

~-1 X~Xm n - 1

C o m p a r i n g with (10) we find t h a t if T ~ 0 , (11) is a l w a y s true.

W e t u r n n o w t o t h e case T > 0 , a n d s t a r t w i t h a few observations. I f (x~ +a~x,) is g 0 , we m u s t h a v e

x,~_ ~ la~l.

T h u s we can a s s u m e t h a t this

' = 1 $~1

s u m is > 0. F r o m (2) it follows t h a t we h a v e either

o r

I f (12) is t r u e

(x~ +atx,) g ~ (alx,+a~)+c

(12)

i = 1 I - 1

(x~ + a~ x~) > ~. (a, xi + a~) + c. (13)

tffil i = l

~ x~_ ~ a~-t-c,

a n d t h u s all t h e n u m b e r s x~ are b o u n d e d . I f on t h e o t h e r h a n d (13) is t r u e we d e d u c e f r o m (8) a n d (7)

( n - 1)x>=n V~-4tl,.

On s q u a r i n g we g e t

2 n - 1

- - X 2"

n ~ t~ _>- 4

W e n o w p r o c e e d e x a c t l y as in t h e case T - 0 . T h e result is

x i~i 4 n 2

z~-xm ,ol z, _-< 2 n - 1 [(~ - 1) (a~ + 1) + l a , I + c].

(14)

ARKIV FOR MATEMATIK, B d 3 nr 17 Combining this i n e q u a l i t y with o u r previous results we h a v e t h a t e v e r y f u n d a - m e n t a l solution of (2) satisfies t h e i n e q u a l i t y

x ~ 4 n 2

x~ < [(n - 1) a 2 + a + n - 1 + c] = C, (15)

XgXm i~1 = 2 n - - 1 where a = m a x l a, l-

T h e following t h e o r e m is an i m m e d i a t e consequence of (15):

T h e o r e m 2.

The number o/di]/erent ]undamental solutions (chains o] solutions) is /inite.

(15) gives at once a n u p p e r b o u n d for x. F u r t h e r m o r e it is possible t o deter- mine explicit b o u n d s for all t h e x~. I n fact, t h e following i n e q u a l i t y is easily d e r i v e d f r o m (4) :

i~=g, m Since, according to (15), t h e p r o d u c t

1 I~I x~

Xt~ Xm i=l

is b o u n d e d t h e same is t r u e for t h e s u m ~ (x~ + a~) 2.

t#g, m I n fact, we h a v e

(x~ + a~) 2 < (C + a) 2 + (n - 3) (a + 1) ~.

t4.u, m

T h u s we h a v e for x, x~, i = 1 , 2 . . . n t h e inequalities

x~_C,

[xi] < V ( C + a ) 2 +

( n -

1 ) ( a + 1 ) ~ + c + a .

E q u a t i o n (2) with c < 0 m a y be t r e a t e d in t h e same w a y as w h e n c > 0. H o w - ever, i h e details are m o r e complicated. I shall r e t u r n t o this case in a following p a p e r .

R E F E R E N C E S

1. ~-~URWITZ, A., U b e r eine Aufgabo d e r u n b o s t i m m t e n Analysis. Math. Werke, Bd. I I , S.

410-421.

2. BARNES, E. S., On t h e Diophantine equation x 2 + y 2 + c = x y z , g. L o n d o n Math. See. 28, 242-244 (1953).

3. MILLS, W. H., A s y s t e m of quadratic Diophantine equations. Pacific J . of iVfath. 3, 209-220 (1953).

4. - - A m e t h o d for solving certain Diophantine equations. Prec. American Math. See. g, 473-475 (1954).

Tryckt den 21 april 1955

Uppsala 1955. Almqvist & Wiksells Boktryekeri AB