A R K I V F O R M A T E M A T I K B a n d 3 n r 47
C o m m u n i c a t e d 14 M a y 1958 b y LENNART CARLESO/~
On homomorphisms and orthogonal systems
B y MATTS ESS~N
1. L e t G be a compact topological group, a n d let D be a subgroup of the group of all continuous homomorphisms T of G onto G. We define a class of functions A~: ] q AD if for a n y two homomorphisms belonging to D
f / (T1 x ) / ~ x) d x = 0, (1.00)
G
whenever / ( T l x ) a n d ](T2x ) are different functions; and if
f /(TI~) / ( T ~ ) d ~ = 1 (1.01)
G
whenever / (T x x) and / (Tz x) are identical. (We h a v e especially f ] ! (x) I S d x = 1).
G
!(x) is continuous. (1.02)
W e can also say t h a t {/(Tx)} is a n orthonormal system.
W e can now formulate the following problem: when does the fact t h a t / ( x ) E AD i m p l y t h a t { (x) is a character or a simple combination of characters.~ We shall t r e a t two special groups. I n the first case, the group G is the topological group dual to the real line under the discrete topology. All functions ] ( x ) E A D are almost periodic, and L e m m a 2 gives the tool we need for the proof of Theorem 1, which is the m a i n result of the paper. I n t h e second case, the group G is the unit circle under t h e usual topology. B y using T h e o r e m 1 we obtain a result for periodic functions w i t h derivatives of all orders (Theorem 2). I n this case we shall also investigate what happens when the orthonormal system { / ( T x)}
is complete (Theorem 3).
2. L e t G be the topological group dual to the real line under t h e discrete topology [1], and let t h e group D consist of those homomorphisms of G, which, when xEG is real, h a v e the form T x = a x (a is a real non-zero number). On the real line, the class of continuous functions on G is identical with the class of almost periodic functions, and we h a v e
+ T
O - T
We can now define this special class, which we shall call A0, in a less a b s t r a c t way. We say t h a t the almost periodic function / ( x ) E A 0, if
M. ESS~N, On homomorphisms and orthogonal systems
M{/(ax)/(bx)}=O, if a * b a n d ab*O,
a n d if M { I/(x)In} = 1.
+oo
a e ~kx
L e t / (x) ~- ~ k •
- o o
(2.00)
(2.01)
Condition (2.00) t h e n implies t h a t
~=t, a,,5+,=O, if a * b a n d a b * O ( P a r s e v a l ' s r e l a t i o n for a l m o s t periodic functions), or simpler
~, a , 5 ~ = O , if u * l a n d . u * O .
+c¢
Condition (2.01) implies t h a t Z [ a k l ~ = l . L e m m a 1. Let / 6 A o. Then M {/(x)} = 0.
(2.02):
(2.03)
Proo]. L e t 2o = 0 . W e o b t a i n f r o m (2.02)
lao]2+ ~' a,a+,=O, if u * l a n d 0.
2~= u2lt
( Z ' m e a n s t h a t t h e s u m is t a k e n o v e r all non-zero indices v a n d # . )
{2~/]tl,}" is a c o u n t a b l e sequence of real n u m b e r s , a n d we can choose a n u m b e r u different f r o m all t h e s e n u m b e r s . T h u s [ % [2 = 0 a n d t h e l e m m a is p r o v e d .
2.1. W e shall n o w show t h a t it is sufficient to t r e a t t h e case, w h e n t h e F o u r i e r series of / ( x ) contains t e r m s with p o s i t i v e frequencies only. W e c a n a l w a y s a s s u m e t h a t this series c o n t a i n s t e r m s w i t h t h e frequencies 2k a n d - 2 k (if t h a t is n o t t h e case, we can insert a new t e r m a(k-)e -i*kx w i t h a(~-)=0). W e can t h e n r e n u m b e r t h e t e r m s in such a w a y t h a t 2 - k = - ~ t , f o r all indices k, a n d t h a t
~t, is positive, if k is positive. L e t us s t u d y the e v e n f u n c t i o n
e l (X) = l (Z) + / ( - - Z)
a n d t h e o d d f u n c t i o n gn (x) = / (x) - / ( - x).
T h e f u n c t i o n gl (x) is t h e n r e p r e s e n t e d b y a cosine series with p o s i t i v e frequen- cies {~tk}~, a n d coefficients {bk}~ °. F r o m (2.02) a n d (2.03), we g e t t h a t
b ~ b , = 0 , if u * l a n d 0, (2.10)
a n d ~ [ b ~ [~ = 4. (2.11)
1
°~ b W e n o w define a n o t h e r f u n c t i o n : L e t hl(x ) h a v e t h e F o u r i e r series ~ k e t~kx.
1
B y (2.10) a n d (2.11), hl(x)/2 E A o. T h e r e is a one-one c o r r e s p o n d e n c e b e t w e e n
ARKIV F6R M).TEMATIK. Bd 3 nr 47 :gl (x) a n d h 1 (x). I n t h e s a m e way, we can s u b s t i t u t e a f u n c t i o n h 2 (x) with F o u r i e r
o¢
series ~ ck e ~kx f o r ga (x), which h a s t h e F o u r i e r sine series ~ ck sin 2k x, such
1 1
~ h a t h a (x)/2 E A 0. L e t us s a y t h a t h (x) belongs t o t h e class As, if h (x) E A 0 a n d if h (x) h a s a F o u r i e r series of t h e f o r m ~ ak e ~akx, w h e r e 2k > 0.
1
I f c o n v e r s e l y we s t a r t f r o m t w o a r b i t r a r y f u n c t i o n s h 1 (x)E A s a n d h 2 (x)E A 1, we o b t a i n t w o n e w f u n c t i o n s gl (x) a n d g2 (x) (even a n d odd, respectively), which a~lded give a function which belongs t o A 0 (from h l ( x ) = a e tnx a n d h 2 ( x ) = e ~mx, w h e r e l a l = l b l = l , we get t h e f u n c t i o n a cos m x + b sin n x , which belongs t o As). I t is e v i d e n t l y sufficient t o s t u d y f u n c t i o n s b e l o n g i n g t o A r
2.2. L e t [ (x)E A s a n d let [ (x) h a v e t h e F o u r i e r series ~ a k e ~k~. T h e corre-
1
s p o n d i n g s u m (2.02), w h e r e all frequencies are positive, can be i n t e r p r e t e d as a c o n v o l u t i o n , w h e r e t h e o p e r a t i o n m u l t i p l i c a t i o n is used i n s t e a d of t h e usual o p e r a t i o n a d d i t i o n . This leads us t o t h e following l e m m a .
L e m m a 2. Let ~ l ak ] be convergent. Let
1
o o
-- ~" a e t~kx / (X) -- ~
1 oo
- - ~ ' a {~ - t y l ° g ~[k
a n d F (i y) - z, k •
1
T h e n / (x) E A s i/ and only i / I F (iy) I ~ 1.
Proo/. I f we a p p l y P a r s e v a l ' s r e l a t i o n t o F ( i y ) e uv a n d F ( i y ) , we o b t a i n i { I F ( i y ) [ ~ e ~ t Y } = ~ a~5~,. (2.20) :Let / ( x ) e A x . T h e n t h e s u m (2.02) is 0 if u * 1, a n d 1 if u = 1. T h u s
M ( I r ( i y ) I~e "y) = 0 , (2.21)
i f t * 0, a n d M { I/~ (i y)]~} = 1. (2.22)
'This gives F ( i y ) ] ~ - - l , a n d t h e necessity of t h e c o n d i t i o n is p r o v e d .
L e t n o w F ( i y ) l ~ - - l . I t follows f r o m (2.20) t h a t t h e s u m (2.02) is 0 e x c e p t w h e n u = 1, a n d t h a t ~ ]ak I ~ = 1. T h e sufficiency is p r o v e d .
1
~" a e i ~ k x
2.3 T h e o r e m 1. Let / (x),~/_, ~ , let ](x) e A s be such that
1
] ak l ~x = M (x) (2.30)
1
is convergent f o r - c ~ < x < c~, and let the sequence {~k}~ at most have one o / t h e points 0 and c~ as accumulation point. Then /(x) consists o/ one oscillation only.
A n assumption that M (x) i s / i n i t e only when x > b or when x < b, is not su//icient to ensure this.
M. ESS~N, On homomorphisms and orthogonal systems
I n t h e proof of T h e o r e m 1, the following t h e o r e m b y H. B o h r is used [2].
L e t the a l m o s t periodic function /(z) have only negative frequencies, a n d suppose t h a t t h e set of frequencies does n o t contain its u p p e r bound. Then ] (z) takes t h e value 0 in every right half plane.
We can assume t h a t all frequencies {~k}~ ° of /(x) either are >/1 or ~< 1.
L e t us first assume t h a t no smallest frequency ;tk exists, a n d t h a t all frequen- cies 2~ are>~ 1. Consider the function
F (z) = ~
ak e -z lo~ ~k.(2.30) implies t h a t this series is absolutely convergent for all z, a n d F (z) is t h u s
o o
an entire function. Since ~ ]a~ ] is convergent, L e m m a 2 gives t h a t [ F (i Y) I - 1.
T h e a l m o s t periodic function F (z) has only negative frequencies, a n d their u p p e r b o u n d is not assumed. I t follows from B o h r ' s t h e o r e m t h a t there exists a z 0 = x 0 + i Y0 such t h a t ~v (z0) = 0. We k n o w t h a t ] F (i y) I ~ 1, t h a t is t o say t h a t t h e values t h a t F ( z ) takes on the i m a g i n a r y axis, are situated on the unit circle.
Schwarz's reflection principle implies t h a t
F ( x o + iyo) F ( - x o + i y o ) = l.
B u t F ( z ) is an entire function, and F ( z o ) = 0 , which gives a contradiction. W e conclude t h a t a smallest frequency exists, e.g. 21. The F o u r i e r coefficient a 1 is different f r o m zero.
Consider G (z) = e zl°~' .F (z). Evidently I G (iy) l - 1, and
Thus G (z) converges uniformly t o a 1, when x--> + co, a n d G (z) is an entire function which is bounded in every right half plane. Suppose t h a t G ( z ) i s n o t identically constant. T h e n there exists a sequence of n u m b e r s {z,*}~ with R e ( z , , ) - * - c o such t h a t I G(z,)l--+co. Schwarz's principle gives
G(x,* + iy,~) G( - x ~ + iy,~) = 1.
This is only possible if a 1 = 0, which is a contradiction. We conclude t h a t G (z) is identically constant, t h a t F ( z ) = a l e -zl°~', and t h a t / ( x ) = a l e ~'~:. The firsb p a r t of the t h e o r e m is proved, when 2~ >/1.
I f ~k~<l for all k, we can consider the sequence {1/~k}F={2~}~ ° and t h e function
F 1 (z) = ~ a~ e -z log a'k,
1
which can be t r e a t e d in exactly the same w a y as F (z) in the previous p a r t of the proof. T h e first p a r t of the theorem follows.
W e prove the second p a r t of the theorem b y constructing a nontrivial func- tion /(x), which belongs t o A 1. We s t a r t f r o m an analytic function with absolute value one on the unit circle. T a k e for instance
.&RKIV FOR MATEMATIK. B d 3 nr 4 7 co
z - - a a+(l_lal2)k~=lzk5~_l, l a l < l .
1 - 5 z =
L e t us p u t z = e - W l ° ¢ 2 ( w = u + i v ) . We get
~ ( w ) = - a + (1 - l a I ~) ~ e ~ - l e - ~ 1 ° ~ . (2.31)
k = l
This series is convergent (even absolutely convergent) for u > (log ]a I)/(log 2) and divergent elsewhere. We also have [ F ( i v ) l ~ 1. Then b y L c m m a 2
o 0 k
! (x) = - a ~ + (1 - ]a 1 2 ) ~ a ~-' ~'~ (2.32)
belongs to A 1. Theorem 1 follows.
3, We shall now investigate the case, when G is the unit circle under t h e usual topology. L e t D be the group of all continuous homomorphisms of G, i.e. T x = n x (mod 2•), where n is an integer. The characters of G have the f o r m e i'~. The function / (x) E A~, if
2 ~
21- f /(nx) f(mx)dx=(~n.m
( 3 . 0 0 ) 0(n and m are non-zero integers), and i f /(x) is a continuous function with period 2~.
3.1. The argument in 2.1 applies in particular t o periodic functions, and it is therefore sufficient to consider functions belonging to A~, which have Fourier series of the form ~ a k e tkx. L e t us calI this class of functions A 2.
1
T h e o r e m 2. Let / (x) E A2, and let ] (x) have derivatives o/all orders/or - oo < x < co.
Then f (x) consists o/ one oscillation only. This conclusion cannot be d r a w n / t o m au assumption o/ the existence o/ a /inite number o/ derivatives.
Proo/. L e t / (x) have the Fourier series ~ ake ik~, and let [ (x) have M periodic
1
derivatives. Then l ak [ <~ U ( i ) / k M for all k. Assume / (x) has derivatives of all orders. Then the series
is convergent for all M. B u t / ( x ) 6 A1, for when n and m are integers, (3.00) implies t h a t
~. a~5~,=O, if n * m and n m * O ,
nv~ml~
and t h a t
1
~t[. ESSEN, On homomorphisms and orthogonal systems If u is irrational, then
(2.02) and (2.03) thus are fulfilled, and ](x)E A r We can use Theorem 1, and t h e first part of the theorem is proved. The function (2.32) belongs to As, has 3 / periodic derivatives if l a l < 2 - M and thus gives the counterexample we need f o r the proof of the second part of the theorem.
3.2. Consider the orthonormal system {/(nx)}~+2_~, which we will call F.
Theorem 3. The system F is complete in L ~ (G) i / a n d only i / / ( x ) = a e ~ + b e -'~, where
lal~+Jb[~= 1
Re(ab) =0.
+t:,o
a e tk~ L e m m a 1 implies t h a t a o = 0 . E x p a n d the function Proo/. L e t / ( x ) ~ ~ k •
{ 1 / 2 ~ ) e ~ in the system F.
~
{Oal
ifl~l~>2"l f,~:~j e~X/(nx) d x = if n = l .
o a - 1 i f n = - l .
Since G is complete, we can use Parseval's formula, which gives
2~
o
:From ~ l ae 12 = 1, it follows t h a t a~ = 0, when I k[~> 2. Thus [ (x)= ae 'x + b e -'x. F r o m
2~
~ /(x)/(-x)dz=O
0
it follows t h a t R e ( a b ) = 0 , and the necessity of the condition is proved. The sufficiency follows from
e ' X = a l ( x ) + ~ l ( - x ) .
R E F E R E N C E S
1. L. H . LOOMIS, A n I n t r o d u c t i o n t o A b s t r a c t H a r m o n i c A n a l y s i s § 41 E. N e w Y o r k , 1953.
2. H . BoHR, Z u r T h e o r i e der F a s t p e r i o d i s c h e n F u n k ~ i o n e n . A c t a M a t h e m a t i c a 47, 273 (1926).
Tryckt den 19 september 1958
Uppsala 1958. Almqvist & Wiksells Boktryckeri AB
A R K I V F O R M A T E M A T I K B a n d 3 n r 4 8 Communiqu6 le 10 septembre 1958
Sur l'6quation x~-t-y" ~=z
P a r TRYCVE NACELL
1. Dirichlet a Ie p r e m i e r m o n t r 6 clue l ' 6 q u a t i o n
x 5 + y~ = z 5 (1)
est impossible en h o m b r e s entiers diff6rents de z6ro. Le b u t de cette N o t e est de m o n t r e r que l ' 6 q u u t i o n (1) est impossible en n o m b r e s enticrs dans le corps q u a d r a t i q u e cngcndr6 p a r V5. Ce r 6 s u l t a t est, bien e n t e n d u , compris dans le th6o- r~me ggn6ral de K u m m e r sur ! ' 6 q u a t i o n de F e r m a t . E n effet, il r6sulte de ce th6or~me que l ' 6 q u a t i o n (1) est impossible d a n s le corps d u quatri6me degr6 engendr6 p a r e ~./5.
2. D a n s le corps K (~/5) u n e base des n o m b r e s entiers est donnde par 1, ½(1 + Y5).
D a n s la suite nous d6signons ce corps p a r K. Le n o m b r e des classes d ' i d 6 a u x est 6gal £ 1. L ' u n i t 6 f o n d a m e n t a l e (ceUe qui est > 1) est le n o m b r e e = ½ ( 1 + ~/5), d o n t la n o r m e est 6gale ~ - 1 . Le n o m b r e ~ = ~.5 est u n n o m b r e premier d a n s le corps.
N o u s allons d 6 m o n t r e r quelques lemmcs sur les unit6s du corps.
L e m m e 1. L a c o n d i t i o n n6cessaire et suffisante p o u r que l'unit6
E = _[_~M ( M = 0 , -+l, + 2 , etc.) (2)
soit congrue £ u n n o m b r e rationnel m o d u l o 5, est que M soit divisible p a r 5.
Ici e = ½ (1 + Vg).
D ~ o n , ~ r . t i o ~ O n a ~ = ½ (3 +
~/~), ~
= 2 + / g , ~ = ½ (7 + 31/~) e t ~ = ½ (11 + + 5 ~ / 5 ) . D o n c e 5 ~ 3 (rood 5). Si E est congrue £ u n h o m b r e r a t i o n n e l m o d u l o 5, et si M = 5 m + r avec 0 ~< r ~< 4, il est 6vident que l'unit6 ~ est aussi congrue& u n n o m b r e r a t i o n n c l m o d u l o 5. D o n c on conclut que r = 0. D ' a u t r e part, si M est divisible p a r 5, on v o i t ais6mcnt que l'unit6 E est congrue £ -+ 3 M/s m o d u l o 5.
Si on exige que l'unit6 E soit congrue ~ -+ 1 m o d u l o 5, il f a u t 6 v i d e m m e n t que M soit divisible p a r 10. Ainsi nous a v o n s aussi
L e m m e 2. P o u r que l'unit6 E, donn6e p a r (2), soit c o n g r u e £ -+ 1 m o d u l o 5, il f a u t et il suffit que M soit divisible p a r 10.
3. N o u s a v o n s aussi bcsoin du lemme s u i v a n t :
L e m m e 3. Soit E u n e unit6 dans K telle q u ' o n ait E - - 1 (mod 5). Si le n o m b r e
T. NAGELL Sur l'gquation x 5 + y5 = z ~
E 5 - 1
H = - - (3)
5 ( E - 1) e s t u n e unit6, o n a n 6 c e s s a i r e m e n t E = 1.
Ddmonstration. H est 6 v i d e m m e n t p o s i t i f e t aussi le n o m b r e eonjug6 H ' . D o n e n o u s a v o n s
H . H ' = + 1 . (4)
V u que E = I ( m o d 5) il r 6 s u l t e d u L e m m e 2 q u e l a n o r m e d e E e s t 6gale + 1. D o n e n o u s a v o n s
E . E ' = + 1 . (5)
I1 s u i t d e (3) e t (5) q u ' o n a
H ' = _E '5 - 1 _ 1 - E 5
5 ( E ' - 1) 5 E 4 ( 1 - E ) "
D e (4) o n a u r a alors
[ E
s - 1"~ z. 1 H H ' = I = \ E _ I ] 2 5 E 4E 5 - 1 5 E 2.
d'ofi - - =
E - 1
O n v o i t a i s 6 m e n t que e e t t e 6 q u a t i o n a d m e t , e n d e h o r s d e l a r a c i n e d o u b l e E = 1, les d e u x r a c i n e s E = ½ ( - 3 ___ V5). C o m m e a u c u n e d e ces r a c i n e s n ' e s t p a s --- 1 ( m o d 5), le l e m m e se t r o u v e d e m o n t r 6 .
4. S o i t ~ = ½ (a + b V5) u n n o m b r e e n t i e r d a n s K qui n ' e s t p a s d i v i s i b l e p a r ~/5.
Ici a e t b s o n t d e s n o m b r e s e n t i e r s r a t i o n n e l s , a n o n d i v i s i b l e p a r 5. A l o r s on v o l t a i s 6 m e n t q u e
a 5
~ 5 _ _ - - ~ _ 7 a a (rood (VS)a).
Or, l a cinqui~me p u i s s a n c e d ' u n n o m b r e e n t i e r r a t i o n n e l , n o n d i v i s i b l e p a r 5, e s t e o n g r u £ ou _+ 1 ou ___7 m o d u l o 25. D o n e n o u s a v o n s le r 6 s u l t a t :
L e m m e 4. Si ~ e s t u n h o m b r e e n t i e r d a n s K qui n ' e s t p a s d i v i s i b l e p a r V5, le h o m b r e ~5 est congru ~ o u + 1 ou + 7 m o d u l o ((g)~.
A j o u t o n s q u ' o n a ~4--__ 1 ( m o d ]/5) p o u r t o u s l e s e n t i e r s ~ n o n d i v i s i b l e s p a r Y5.
5. U n e cons6quence i m m 6 d i a t e d u L e m m e 4 est le
L e m m e 5. Si l ' 6 q u a t i o n x 5 _ y S = z 5 ( x y z # 0) (6) est r 6 s o l u b l e en h o m b r e s e n t i e r s x, y, z d a n s K , il f a u t que l ' u n d e s n o m b r e s x, y, z s o i t d i v i s i b l e p a r V5.
A l l K I V FOR M A T E M A T I K . B d 3 n r 4 8
Ainsi on p e u t supposer dans (6) que z e s t divisible p a r V5. hTous p o u v o n s aussi supposer que les n o m b r e s x, y, z sont premiers entre eux d e u x & deux.
6. A u lieu de l'~quation (6) nous consid6rons l ' 6 q u a t i o n plus g6n6rale & cinq n o m b r e s i n c o n n u s
x 5 - y~ = E . 2 ~ z s, (7)
o~ nous a v o n s pos~ 2 = }/5.
N o u s allons m o n t r e r que cette 6 q u a t i o n est impossible q u a n d les ineonnus x, y, z, E e t / z satisfont a u x conditions suivantes : # est un n o m b r e entier ration- n e l > / 0 ; E est u n e unit6 dans K ; x, y e t z sont des n o m b r e s entiers dans K tels clue x y z ~ O .
S u p p o s o n s a u contraire que l ' g q u a t i o n (7) soit rdsoluble et ddsignons u n e solu- t i o n p a r Ix, y, z, E , ~]. :Nous supposons q u e (x, y) = (x, z) = (y, z) = (x, 2) = (y, 2) = 1.
L e t e r m e ~ g a u c h e dans (7), x S - y 5, est le p r o d u i t des trois facteurs x - y , 2 x y + ½ ( 2 + l ) ( x - y ) ~, 2 x y + ½ ( 2 - 1 ) ( x - y ) ~.
Le plus g r a n d e o m m u n diviseur de d e u x quelconques de ces h o m b r e s est 6videm- m e n t 6gal £ 2. P a r c o n s e q u e n t on conclut de (7) q u ' o n doit avoir
- y = ( s )
x y + - ~ - (x - y)~ = E1 x[, 2 - 1
(9)
off x 1, Yl, Zl s o n t des n o m b r e s entiers dans K tels que (Xl, Yl) : (Xl' Z1) = (YI, Zl) : (Xl' 2) : (Yl, 2) = 1.
E0, E1 et E2 s o n t des unit6s dans K. E n gliminant x et y des ~quations (8) et (9) on a u r a
5 5 1
-t2J 0 " IL. ~ ] .
E1 xl - E 2 y , = ] ( x - y ) 2 = ~ 2 ~ + 2 ~ .10
d'ofl x~ - E3 y~ = E4 2 S+2~ zl °, (10)
off E 3 et E 4 sont des unit~s dans K. E n a p p l i q u a n t lc L e m m e 4 ~ l ' ~ q u a t i o n (10) nous a u r o n s
E s - + _ l ou ~-___2(mod22).
D'apr~s le L e m m e 1 l'unit6 Ea eat d o n e u n e einqui~me puissance d ' u n e unit6 d a n s K. Si n o u s posons E 3 - E 5 - 5, x l = u , E s y l = v et z~=w, l ' 6 q u a t i o n (10) p e n t s'~crire
u 5 - v 5 = E 4 2 ~+2" w 5. (11)
Ainsi, d ' u n e solution Ix, y, z, E, #] de l'~quation (7) nous a v o n s o b t e n u u n e a u t r e solution [u, v, w, E4, 2 # ] . Ces solutions s o n t li6es p a r la relation
T. NAGELL Sur l'~quation x 5 + y5 = z 5
z = u v V w E g 1. (12)
E n p r o c ~ d a n t d e ta m ~ m e m a n i ~ r e a v e c l ' 4 q u a t i o n (11) n o u s a u r o n s u n e t r o i - si~me ~ q u a t i o n
5 5=E6~5+4~,wSl, (13)
U 1 - - V 1
0~1 Ul, Vl, W 1 s o n t des n o m b r e s e n t i e r s d a n s K tels que
( U l , Vl) = ( U l , W l ) = (Vl, W l ) = ( U l , ~ ) = (Vl, ~) = 1.
E 6 e s t u n e u n i t 4 d a n s K .
N o u s a v o n s a i n s i o b t e n u u n e t r o i s i ~ m e s o l u t i o n [ul, vl, Wl, Ee, 41u ] d e l ' d q u a t i o n (7). E n t r e les s o l u t i o n s s u b s i s t e l a r e l a t i o n
W = 71,1 V 1 V g 1 E 7 , ( 1 4 )
off Ev est u n e unit4.
P a r m i r o u t e s les s o l u t i o n s [x, y, z, E , ~u] d e l ' ~ q u a t i o n (7) il y a a u m o i n s u n e a v e c la p r o p r i ~ t ~ s u i v a n t e : L e n o m b r e des f a c t e u r s p r e m i e r s non-associ~s d u h o m b r e z e s t m i n i m u m , e v e n t u e l l e m e n t = 0. S u p p o s o n s m a i n t e n a n t q u e n o t r e s o l u t i o n i n i t i a l e [ x , y , z , E , # ] s a t i s f a i t g c e t t e c o n d i t i o n m i n i m a l e . Alors il r d s u l t e d e (12) e t d e (14) que les n o m b r e s u, v e t u 1, v I s o n t des unitds.
E n t r e les n o m b r e s u, v, u I e t v 1 s u b s i s t e n t les r e l a t i o n s s u i v a n t e s , a n a l o g u e s a u x r e l a t i o n s (8) e t (9) :
u - v = Es" ~3+2~ z~, (15)
i + l
u v + ~ (u - v)~ = E~ u~,
~ t - 1
u v + - - ~ (u - v)2 = Elo v~,
(16)
oh Es, E 9 e t El0 s o n t d e s unit~s d a n s K . E n m u l t i p l i a n t les d e u x r e l a t i o n s (16)
o n a u r a
~ 5 __ V 5
E9 El0 (ul Vl) 5.
5 (u - v)
I c i le t e r m e g d r o i t e est u n e u n i t & Si n o u s p o s o n s E = u / v , le n o m b r e E est u n e u n i t ~ ¢ 1 . D ' a p r ~ s (15) on a E - - 1 ( m o d 5). I1 en r d s u l t e que le n o m b r e
E 5 - 1
H = - - 5 ( E - 1)
d o i t fitre u n e unit~. Or d ' a p r ~ s le L e m m e 3 c ' e s t i m p o s s i b l e q u a n d E ~ 1 ( m o d 5) e t E # I .
N o t r e t h 4 o r ~ m e s u r l ' ~ q u a t i o n (7) se t r o u v e a i n s i d e m o n t r d .
T r y c k t d e n 14 n o v e m b e r 1958
Uppsala 1958. Almqvist & Wiksells Boktryckeri AB
