Some Congruence Properties of the Partition Function

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S,r',ll- (;tll'lCiilri1:tlt, i'R(]Pili i I i:.ll '{l|E 01; Pl|i:'t'i't I trlt lrilt0 i1 trN bV Ir::ancis

Su]>rlit:j-e,i

l-c)r t--ite deg::ce

G.

Ga;'vtrn

of

i,i*t/.c;r- r; i Sci.,trtct, Urr j-rie'r:s;j,t','

of

liev;

ljout-lr

1i.-l1es,

-19E2 .

i.

coli'MNT:i

l.llTROD'!lCTIOi.J

CHI\PIILR I

A simplc

l)r-ocif of

con(-rl:ucn(ic5

CI]APlll:R tI

Atkints

of

C H A l ' ] T E RI I I

ii

for

l.iat-sorr's 1,nrt i t.ioit

r)c\vcr'9 0f

7.

con,rl-ui-:rru€s noilulu

I,ari--it.ion

])o\t'ers

2l

5.

Some new congl:ltenccs of modufo po\{el:s of

7.

C i " l A p ' I ' i J rl iV

Congruences

p - - K, ( n )

C I { A P T E RV

An element_ar:y proof

for

aud some furtlrer

BIBLIOGRI\,IIHY

of

Lhe par'.ilion

funct.ion

42

51

p(l-1n+-6) :

0

mod }l

93

ider:rt-i1-ies.

L26

l- -1, .

Thtnrf I ilY lart1l-r)\]

It't ti-riS t-hCSiS v;l ' il r e p a r L i

involving

Obta-in

I . - ir : i

funcLiorr

Recent"J-y Ilirschh:rril L):nrrrrr-i

:-,r'-.,

2 . , - . - i 6 6 1 rur\'{-r .. ' evrr_)ee

rf vr tr

i

r

r r 6 r . ; c r - .5r

6/'!1 L

IrL/\vcr

hov; t-heir: il'iethoi.s car) be extenclcd for

1 > c i ' , ' e : r 'a: f: . J "

fonva::d classi-ca1

than

lllose

of

iderrt.iti.es

follorv

'r-rrri

r,'r.

Inodul.o po/rcrs congrucnc:cs.

of

Iiuler:

i -u

and llunt of

5.

In

In

rur Jc ,i ! l \nJ n

thild

l.le also

olornnrri:yrr

drle tci Atki.n.

\.,.e lijlo,r-

oongi:L.Lences

f ormula.:

six

we obtain

analogous

fl:nci--ions

l^le gi.r.e a det*rle

on

linll'nr.a

congruencesi

thr:I

(1.1..e)

d .

M = (m ""i

The first

)

i'i

.

-

ts = (b.

Itf

t L r - .

.

are dcfineil

. ) . r t ) r>\ Lr

L p ,

. f

trl-r]-i-l

is

i>'l

0

{ foB'

a n d

Irl

r.rhere

^ r . . . . 1

z

L 1

I,

(r.r.B)

Here:

. rX,

L T L

-

Irl

- --

.

l l t - - .

by

_

bl-+.lr1+l

defi,ned as follovrs:

f iv'e rows; of

J

V

0

0

0

0

o

0

0

0

0

)

0

0

j

(1.r.i0)

and

for

r

(f .1.ll)

2x5

5-

9

{ 3x5-'

4

) 22x5"

I

4x53 -

b,

i

n

l.rl

Thei, then pr()veo **,i

= 25m. -

.l_*Irl-I

A Y E -

8x5-

and for

I,l.

m.

0

a, 0

l--/.t)-L

nod 5r0

i{at,sor.r also

prc.verd that

( 1 . . 1. 1 2 )

' . - ' (a'/ i"ir r - r i r ^ 1 n--l ' ) 5'' LI,

if

J

-+ 25rn,-.

( I . I . 1) by showing for is

J. :

5

! . 2

0

t

.+ 15nr. ^

]*Jrl--L

,*

5rn.

J_-4rl-I

that j > 1. odd and aL icasi: at

O

-

mod 5'"

3,

then

.+ m . L

J r J

?

chor'iltr [ 7 -l cupt-a [ 9,1

noticed

oi = 3.

f.r-j-is for

fn

fact_ frcm

v.'e ]ra..,-e P ( i.r)

- p(243)

wli:Lch j-s c1,j-r.j-si1:l.er bir

= r 3 3 9 ' l 8 2 5 9 3 4 4 E E B,

j2

but

a p l ) r o p r - r _ a t e r r o C . | f : L c : a r t _ i ocnf .-2R--1 *n

(t 11 , , 1 . 1 3 )

(1,r.2)

t]rat

p(7''''

not

73 .

I,,tatson [16 ] proved

.lfiat if

(I .J.2) , viz.

3 > f

then

mod 7P

-r

ancl

p 1 7 : 2 8 , ,* ) . ^ ^ ) = o

Vlal--son also

!,rovecl

(l .1.14)

1-B ) D - t1' ) 2a. r , ( 7 ' - u n - r -) . " o - 4 . ' 7 " = -p ( 7 ' t ' n * A ^ . . zp 2.b

*o,t 7flt1.

zp

that

?, )- L,

:i f

thcn

2. t

..e__l -)

. = p ( 1 2 3 n n t r r B - l 2 ' ? ' - 1 )= o In

this

1 l r(\ \! _ r-._L.r))

n

p. (, /- c In + / . c i ) q I n>0 f

on the nodul-ar

trll

.

but

e : q t L a t - i - o uo f

we de,r'ive it.

llirschhc>rn and I{unt.

for for

t c l i iI . t . 7 )

for

s e v e r - r t - i ro r d e r . uslng

of

calculatj,ng

functjons

in

I { a t s < . r r - r ' s ; > r o o f sr e 1 . y

lle af so neecl tfre moclular

tJre clerncrrl-ary techniques

our proof

for

1,

fol-lorv earsily.

(1 .1.13)

The main-result

an algo::itirnt

the generating

0 )

ior

and (f .l-.l4)

The remeLinder of

conLalns

analogous

mod76+1.

furrcLions

fi:omvrhiclr (f .I.I3)

equration

jcler-rtities

ch;rpt-e::we e:;tabfish

ther gener:aling

t]re

R

+,\r,,_.,) = 0 LtJ

by

of the

(I.t.15)

is

analoEour; to

tl-ris chaptcr, coef f.icients

.

c>f o.

!{e carry

Kolberq that-- of

starteC br3low, in

out

tlte

fornul.ee

tLrese c;rlculati.ons

a = I,2.

The main re:sult ' I H t r O R I l t (" 1l . 1 . 1 6 )

of Tf

th:Ls chag'1sr i5 0>

1, . ) Lr i > l

I n>O

r , ( 7 0 n+ ) ) . , ^ Lr

x

j n v t L

q-

i - T*

'l 4 i - 1* r_ J:_-' . _( _o_ _' _) - " -:' r' ' il ( . r, trl ].rl?_r

( 1. 1 . 1 8 )

ivhere

ui,

lv{= (m. *) . :\ r Lt r, )

The fi rst

j

ocld,

l-'

1

I

I fo B, Here

t

B =

(b. : rL l r l * ) , 1 . ,t )r >

= nt4i,i*i

is

\1 even.

'

bi,

11

arcl

dcfj.rrcd by

= *4i..r-r-,i+jI

clefined. as follov;s:

J..L

seven ror,,'s of

j

are

q

(1.1.19)

l"r

f.J \

\D O

r

c,

|

.

;

F

x

.

€

x {

t !

! N

}J

F,)

Lo o\ x

)


c

)

O i

x

{

t

)

! L

a

x o

X

{

r

!

N

€

!

fJ

J

x

!

o

N'

i

!

u

X

!

(

u N cl 5

t! . X c 5

! r

@

!

Ar

\

@ L, X

!

U X

X

!

J

u

s

X

\J

\ o\ X !

o\

N

v)

N

o

andfor

i>

n

tni,l=0,

i > B

for

*i L t . .

'/*i-,,j-1-t

(1'1'20)

,

= 0

i > g, j > 3,

a . r r df o r

3 l t * r - r , j - r - n o n * r - r - , j - i . ** i - r , j . - 2 * 7 * i - 6,j-2

"t,,

: !11*i-s 49*r-n 147rur-a, j - r * t n t n ti * 2 , f - 2 + 3 4 3 o i - r, ) - 2 . ,J^2t ,)-.24 Thecase

3=l

( 1 . 1. 2 r )

.., "d)1

(1.1.3)

] pro.red

If

method of modula:r equations

met-hoci of

pr,:of

rc:l-ies

ntodul;rr: functior,s

n(q)o

for is

clencral

It

cr

rrot sufficient.

on the behaviour

and the Fourier

l +1. { d L A I l l

series

appears -

| ^ 5

of

bl

Ilt " t

IJ

d l

l c

as well- as I':Lnets i B ] ntoiular

entirc

l

:

a,b,crd

€'iZ,

el

usirrg trlatson's

(t

q49t-")

]-

n=0 nroil 7

trZO mod 7

.

(r

Y

J

,a (1. -

q

7n.B )

t n

il

(l

il

-

6 "

7n. 1

t

l

n>l

- .

I f

r (1 - o,7") l1 n) ln:0 nicd 7

n>f

n r. rr) l

(r - e7t)

n

tEJ

[

{1 - r.,r6nnn;

E( q ' )" ,4 ci-

4q

(r - q="')

E( s ' - )

n21

1.3.

The in.ain r:esult of

relies in

thj,s section

on the n.duLr-ar equtrtion

lVat-'sonts.pape:r but

I{e now intr.ocluce

of

is

seve'th

lemrna (1.3.1) . (1.3.14),

order

r ^ r r ' r i c r rv , , e o b t - a i n b y a n e r e m e n l a r y

the

oi_rerators

H. , 0 < i < 6

of

powe:: of

porvers of q

is

L e m m a( 1 . 3 . 1 )

q

congruent

ancl simply to

i

pick

modulo

vrh.rch appeai:s

metliod.

v i hi c h,act o n a

l-

series

Ou:: proof

out

ihose

7.

set

terms

in

i.ihich the

H = iio.

.For

j

(1.3.2)

r t t -

- ' \

r1\!

)

-:

\.

i

ilr

--t

m . . T -

r - ,J

- 7 4

(1.3.3)

and ttre

v"'here

j ,j

lrle lcave

6 ( q ) = ___Elq) ,t2 ,, (q49) ar:e

r (s)

--i*-x ( q ) q

4q 4E(q--)

def j.ne:C b1r ( 1 . 1 . 1 9 ) a n d ( 1 . 1 . 2 0 ) "

tirc grr:oof of

L r : u u n a .( l . 3 . 1 )

conseq,itcnce r,\'e h.:rv€: i_he fo,r-lowinq

Lcmma.

ti1l.

lat_er .

n>

dtl

i-n;'neCiatc

10. Lemma (1 . 3-:11..

l.oi:

i ) I ,

u ' r ,\ er - 4 i ,

and

r

a. . T*1-l r, J

/ i>l

uB'(4j+])r

- i- _ i

|

rr\1,

b..

/,

! | J*

i>l

,

T

J

wher:e the

u,i,

l.t

Broj,f_.

is

has no t€:rnrs of

lri, j , j

are

e;isy to

c l e fi n e d b y

check

C::gree

i

that

(1 .l.1g)

n(q-4i)

o:: l_ess.

as a polynor.ial

I

jlr

ai itL

and

s;imiltrrJ_y

to

H F rr r (\ L

-* ' 4 i , i

r-j

i>t1

-i :

= it i r j

fir;

r f , r r ;r ,j .

rr(E-4i)=

We can argue

T*1

So by Lernma(-l_.3.1)

n ( g - 4 i )= i . : . r - t - j 1'rl Therefore

'n

I

6i . r-i-l " |)

I j>l

shorr, that

(4i+1) ., )

-=

r

oi.,j

t

_i_ j

a3', Tn order

to

deri'e

results.

the modul-ar: equation

Follovring

(l 11 ' J1 ' 5- \)

Kolberg

I ltl

- )

Q = - n-'Qo,

v/e first

need some 1:rel-ini.arv

vre clefine

B = q-ler

.

and

.r = q- 3-0' 5.

Froln Lcmma (1. 2.I ) r.,e have

(r'3'6)

{(")=-t:%. q

Frorn ( I .2 .2)

E(q--)

;Lnd (I .2 .3)

= s - 2 9 0 - n - t Q r -I - + e 3 Q u = - ( a + B * y + l ) .

r ^ z eo b t a i n

cr?'2+ cr2 + y = g g \ ' * ' 8 2 * o = o 1 1

?

? \

yo,2 + y2 + B = o cti?,Y= .t-

t

l

T ^ +

(l

?

' l 'hdh -.- \- tL

a\

I

Y r = o 3 B, ( 1 , . 3 . 7 ) , w ee a s i l y

l - .. py

Y2 = B3y

find

and

YrY" = -Y, t

.

= -

'

)

f

'

f , v- ) . 'vL

f

Y.. -

1,

J

' l

-

o, = y'o

/

1

=

q,

* J.r

-

v

-

'

l

tr

= y- ^z Y3 + 1'

frY

YOl

=

V . ' l

-

J

-

rt ,

r:I

P

= - y- Z.Z + y " - y . 4

The foll_owinrJ

Leruna is

r

5 .20 , 5 . 2l- and 5 . 14 of

Iemmafi .3.9) . ------:_

_ Ll t

y , y ^ ) r _ = =f ,

L

2 3 Y 2 -t ' Y c t

I

= - \/ J 1

_ f,

y

\/ ) 1

f'

l ! '

I

'.1

I

- - Y : t * ) ' r - Y . z

Kolberg ,s pap,er..

YI * YZ * y3 = - t - B, YIYz * Yzy3 * y3yl = T + 5, y1y2y3

wher:e

T

is

clefined

i n ( 1 . 3 . 3 ) a n c lt h e

are define0

x(q7)2,

I o . I 7 n + - 6q)t = - t n ( q 7) 3 ,

Ir? J

- - 4 1 ( q ) n- r o n ( q l ) 4 n L> -( )t n , 7 n + 1 . ) q t t and

= 4 e q n ( q 7 )6 ,.,i,,t,u(7n+5)qn

Sirnilarly

'15. Lr:mma(1.3 'l'1i-.'r2 .= wlte::e pt"€.

t.\

J

(The r'rocrular Ecluation of scve'ih

('/E3 + tt;{2 + 49i,)r + E7 + tE6 n zr{5 + T

a.n 3 - 't. ,J j : I - l t .1 ) r - j - 1 * iir

I ( * lr _t t. ), . + T r n l

j > l

i - 6 , J

+ 2 l , r n | _ _ . , n n r ^ i _ n , , *1 4 7 r r r l - r , --l-: .+ -?43rn1 j-' _ 2 , i l 3 4 3 n r l _ . ., , ) :

1 -

=

I ( 7 rrn- jr /

j>3

H€)nce, for

.+. 35*j--r, j - . r * n n * j . * r ,j - 1 *

> g

i

r - J r l - .

I

tt*i-r,

,)-2+

r nl rrl

Be:[ore proving

*i,

for

j

ou:: main

Len,",g__LL_3.t1) .

J

-

.l43mj 1 | -- | '

A

>

147m1-, ..,-?4 every

i, j

> l

.

tne

reciprocal

modulo

7

.Y

of

i l , 4I s a t i s f i e s

1

; 7 0 + l o , dJ-l

* 7 G + l o ,

{ ;

D rnzr

|

.

€

A

0

=

{

1 'r4 ( 1 7 x 7 0 *

24xa = 1 mod Iecurrence.

an rnteger

It

.]

l l

lr4 is

n,l

L ) t

l

i

sincre this

. ^/ ) t - j

- r )a -

Theorenr rye nee.J one more f emroa.

^1 0 , ,

and for:

+

j*r.n'9*i-r, j-f*

+ n9nti-o

Therefore

n

L-t,J_Z

- i > ?

m j - t = 7 m l - + - t J

- l r

l-l

(22x70+1,),

wirich j.s easily

satj-s f ies shown that

O

this

even,

lo

'

7o

lu

and

satisfies

I R

1:-:1

We a::renow in

venience

w{r write

a position

rhe

theorern in

the

r

fori-ov.,:i-.g equi-valent

|

t

*o,

rr),

(1.4.1)

i n>O

T

i

Fot: con_ form

i -tt' T' g-'t t l/s, n (q7 ,,, ,

and

E

are define

"o,i

Tj ' q - a i - ' l / r t

(l.3.3)

in

r(q4e),

.

We have

= *-*-q1I P(")q''''' -_!ti (q)

n)o P'clcing

out

5 m c > d7 ,

those

,J2 E (q49)

terms

r n , eh a v e b y

in

wh'ch

the porver of

q

is

I

or?n + E\^7n _

r

!

I

Novr

T r \(yn r / ' 7 ), =

,. qgr-2

q'r(nnn) - - -n=(;q- )- 4 '

oB 1 - -4 ,r .( l- _4 .9i ,_4_ _ l _ _ - 1 e4 , n r- _ l I s E(q7)4

q E(q')'t

(r.4.2)

lr-L

" '

n2o

( q r. ! - t ( n t / rl = r 6 - . 4

So wc have

I pr'n + s)qn= :g-g:1*- lg'r'-r:

n>o

q E(q7)

Subst j_tutj ng

T '= q

we obLain

cong::ue't

(1.3.16)

= = +c+-= :t1t_gq:, I^ nrTn+ 5)n7n+5 n)o q, (nnt) ;tilq--_ "

or

cx

ocrd,

o

even,

p f T c t r-, r I ) . , n = ttla

where

,ltheorem (1.f .16).

to prove

'

7 4 "tQ,l . E(qoe)a

Ramanujan's result:

and

g-r

- q2 {q' ) E(q)

to

LV.

I prTn + 5)qL n)o which is

thr: ca.se

o = l.

of

7 E!dl' n(r:)r,

on

' - . n-= I -n t 7 0 n- + .'l( ") q ,-,>o out

thol;e

t-erms in

t t

l]lreorerri (1.1.16) .

We now p::oceed by inductjon

Picki.g

'1- . +a) e a . l q t l ;;il

ivnich

o

Suppose

. \' tr3.,

"o,i

o

is

ocid and

- i . - . 'a- it_ , _ . 7 . t- { /q n(q').

the power: of

q

is

co''.

seven rov/s

v I

2 z

t _ ^ ( U I L__--__ . 1 I

O

5 l o o

c c

I_ 5

U

I

.

m

I

:

1 4

@

0

2

'

@

0

2

3

5

7

@

0

2

4

6

1

6

6

oo

@

Def ine

r

6 4

-

I

2

o

t). l._s

0

I

l

-

1 a

2

0

5

3

2

N 9

- 7 -

6 6

7

3

5

.

Frbm (1.1.20) it

t1

j-r)

t6

by ItJ

l_B )o-:rc

1*

= []u(7j -- 2i - r")l

1 . -L(]

1B

20

aa .z

ZJ

s l_1

I2

1 A

1 a I O

18

19

) 1

) "

I

10

12

L4

15

T7

19

a l .J-

Z,a

l0

11

13

t5

1 1

IB

20

2'),

11

13

I4

1 r IO

1B

20

2)l_

l0

L2

L4

t6

I7 l r g

2l

I7

2Ct I I

7 6

10

L2

a.ndfo:l i>7,

foil-owsthat for

v(ni, ') ) rnin {u(*r-a,

13 14

15

2 l l : s6 Observethat:lori l z i - : l r,-J

Proof.

1l

4

4

From { 1 .5 . 1) v r e :h a v e

u(tr,,j) = v(*4j.,i*j) 21-1(!-l jl

u (o r,

i - 1r t

v(b; .) >[]:*t t J

Jr

j ) = t'(*4 i n .,i *j )

ei_-*r

,

;

) 1,7( i + - i1 :JJai

, J+-----j:______r: r .- i

::

[

+ tL_:+

.-'l

A a

= tZr *__l__ =

L e n m a( 1 . 5 . 3 J.

t'(*r-,1)= 1, and for

- 2

't"t,r'

B} T, ' ': v ( x ^ ^ - ;\' 2 r. -R + L 1 a f JJ=-o -) pt :t J

l ,

4

V(x-^

,) "pp ', LL 1 r )

F rn.r

\ 2

r, pe

-: *

f)

+

,[

'

-

+-

'J A

t.

F

L " l !I

(7, Agt,O, 0,

...),

so

V(xr ! t J -

tr{e }iave

X ^ ' = z r f

T

x'1 .

L i >'l

a.

Ltr

-

/d.

l - r J

,,*r,r, - 2

l,

=

.)

49a^

+ r t )

z r f

cn

v ( x ^ . ) ) mintr + z t ) ) min{r + =

min{z

r r /^ v\o- J

l

'

'

' A

l

l - I. - ' , J A

= 2 4. f l-i__l '

'

\

rr)

4

J

-

r

t

^ Z

ll

a

J'

.>

o I t

t

z

rrr/r'l V\o.r ,) | '-rJ

,

t

t

t

, r / t 1 - .l - + _ - A

-

l

J

r]

J

r]

as reclui. r:ed .

Not,vsupposr:

82t

and

v(xz8 > ( [i + ]) + 1Jl_- - o*1 ,i) I'l eh a ve

*2g*1, = bi, j 3 ,1, "r g, i.

SO

V(x^.,,"

z t . ) t r .. t) r

)mi1

i>t

{. .' ( * r g , r )

* u(or,jr]

+t z i - _--li 1 1

= T l li , s . , , . t r y l t:r

(9 + r) * 171-: 3 1, FinaIIy

supp()se

B2t

as resuir:ec1.

and

v ( * z B * r > ( B+ 1 ) * 1 f f ,i) we Lrave

*zg+z,i=

,1, "ru*t,i

1

ar.,j

u ( ' * r g * 2 ,t3T)l l { ' ( " z * * r , : -' )r v ( o . , . ) } )min {tB*l

+t1ii}l

+tZi--+--J11

i>1

= (B + r) n t7t;-2 I = (B + 2) + 1T;--9- 1 , Lemna (1.5.3)

follows

as required.

by incluction.

T h e o r e m( 1 . 5 . 4 ) .

For

6 ), 1, p(72$-Ir, * ^r3-.,-, r tr noci 78 , l i t 2 8 r . r ) , , r U )= o m o d 7 3 + 1

21. Itr:om Theo::ern (1.1.16)

-

f Q

ai-r ^ i - 1 E( q 7 1

l

P rJ " - ' n

)

vre have

+ I ^ . , )s n = zp-l.

n)0

t

L

L

"

.i>1

^at -.P

.

1 L

9

t t

__

1,1

E (cr)

By tenma (l.5.3)

V(x" .) 2t

andfor

L t L

V(x..

.

zl>- L t I

OT,

ft

X ^ ^ zp-r,r

foll-ows

B>2,

2B+tl]4__l>B-

.)

= 0

mod 78,

for

B ) t.

)Q-1

tl)at

P(/

n + ^r*-r)

=o

m o c l7 8

Similarty, e

)

Q

f

Z P ( / n + n)0 and

A " ^ r ' l

\

t d Y

n

I

,I

OT,

x^^

)Q

P(\tfrn *\rf,

1 1 -

v

tYrL

-

E(qr

h

A

-

4rl.I

)-Bt. I

)

Q t t

mo(I

/

n

+ l ^ ^ ) = o

m

n

- F. \ z B -2 . z2' rB' --tt) = p ( t 2) e 3 n* \ r B - f B * t )

ZF'L

Theogs4(I.5.5)

=0

I l '

')ot r (a

^i-1

.

i>1

v(x^. .) ) ( 8 + r ) + zF

trl( /

*

)R

zp

o

R+'l d l

.

P.}'-r

For

P

,t.l2B-I) = P

-

L

,

? R-'l

(

= o

P,+'t mc)d / .

Proof.

From ( 1 . 4 . f )

i

we have

D- . f . e t/- 2 3 - "n1 + t r r u - r ) u "= trl,

n20 rf

we: pick

k mod 7 r

)

n)0

out

those

terms

in

r.;hich the

-'a;

"ru-r,i

i t t g - 4 t l/ s ' . a ( q 7 )

povrer of

q

J.s co:rg::r.lci-ri:to

we L.trve /t 3

A

airai)

r-esults for the

namely P(1,25n+ 99) = 89 x 25 p(5n + 4) mod 56

and

P(625n +' 599) = 3339 x 25 p (25n -r 24) mod 5l'0 In

this

chapter

(2.1.3)

we ob1-aj-n the

followinq

2474) = 240839x 25 p(f25n

p(3125n +

P(15625n 'r L4974) p(78l25n In

f act,

of

5.

a>

I

thcr: Atkir', and if

+ 61849)

s;imilar

[ 2]

has stated is

the

=

::eleLtions

+ 99) *od 513,

2 1 9 3 9 6 4 x 2 5 p ( 6 2 5 n + 5 9 9 ) n ' o . 15 l ' 7 ,

= 2 5 6 ' 1 0 0 2 1 4x 2 5 p ( 3 1 2 5 n +

exist

6^

new cotlqrllence

congrlicnce

r c l a t -j o n s f o r

tlre follo''.ing

result

2474) *oc1 520. all

higlte:: irow'ers

without.

reciptoczrl- nroclu.l.o 5*

of

24,

proof . then

If thcre

L,I,

ovi lact power of

thc

.

lrossible.

5 diviclin L ,

.) 2a+. ['r(si-5)],

o,

odd,

cx

even'

0., I

V(x- .) )o O,,I

r.rhora

] -h o

.r ', 0 ,

i

are

- 4)]'

+ [:t5i

( 1 .I

def -l.rted by

. B) .

Lemma (2.2. j)_.

v ( : r - . ) > ' I L r ( -s j- i - 1 ) ] , - L r ] where the

a.

arl

v(b. L t J

Lt)

Folloiv:Lng Ai:kirr and O'nrien [ 3 ] (2'2'3)

68,r,i

(1.1.9)

are clefined by

., b.

) 2lL,(sj - i - 2)1, -

(f .I.iI)

.

we define

= *2[i*],i x2B-],J - x23-t,i x213+1,1

and

(2'2'4)

x2f,t-2,) for ' ' 8 , i , . . 1= * z ; + z , L x z 1 j - * z B ,!

where the

K

-cx

are defined

(l-.1.8).

in

We have

and tB,i,j = - tB,j,i

6 8 , . t , j= lu,r,' so that 6..

R i P t L t L

= en l

l


1--2t3+.2,): xjt

=u]i,

G z t T + z , k* 2 8 , 9 -- * 2 g , k x 2 g + 2 , g ) b r . b r , , ) ,i ,1,

= 1 , Xlt

\

x2g+3,i

t o , - o b , - . b n

I

FrKrr

Cf

Kra

x'rJ

Si m i l a r l y tB,i,

i

= x2B,+2,i *28

,)

x2B,t x2g+2,1

= [ u I r ' ' z B * r , ki l k , r ) [ n ] , - " , g - r , 9 u- u ,j ) - ( o I r " r o - 1 , k a k , r ) ( n l r " r B + r , ! , aji), =

( * r , l * t , kx 2 B - r - * 2 8 - r , k x 2 g + L ,u k , t Q . , j t) i ,L oi, ul,

= orlr.ulr L e r r n a( 2 . 2 . 6 t .

o I uk,i t[, I B,l t ) f 3- 2 . t a t j r u(r8, i,j) Proof .

From ilirsch.irorn

If I: N o r v

6 ,

t t z r l

and Hunt

l>-g, + s 5.1,

[0],

for

i + i ) :.

we have

= =( 5 , 0 , 0 , 0 , . . . ) a n d = =( 1 3 5 3 8 3 9* 5 3 , = = - 6 .

1 8 8 5 0 2 6 2 i 2x 5 6 , . . . )

^ = x . , x ^ ^

J , 1 r 1 .

I r I

5 r l

s o

' ( 6 r , r , 1 ) ' = v ( 6 , - , r - , r=) v ( x r , t ) + v ( x r , r ) = I + 6 = 7 '

31.

i > 2

For (2 .2.I\

-i .l =

6,

'

6., , L r L

I r r r *

i

r J

= X.r X, L r Lr J

Therefo::e

;.

from Lerma

r )

we have

' ( 6 r , j , r )= v ( 6 r , r , j )2 a + I l r r l r If

i>I

and

j>1Ur,r,j.=O

hencefor i+i):

v ( 6 rr, , j ) r - a + [ l q a j H tt^,, r\uw

'

] = : - I]L#

I.

^,,^'^^^^ DulrPu>c

u ( o 6 , r , r )= 9 9 - 2 15(i+i) u ( 6 g , r , ) r r ' s B *- 6

and

From Lenrma (2 .2.5)

e^ . . = Frrrl

I xir

-el

rot i+ j):.

we have

a^ I O^ . ^ a. k,1 L,J tir 5,K,f

and

6^ . ^ = 0 FJ,K,f,

for

k = l,

s o b y L e m m a( 2 . 2 . 2 ) v(e^

Frrrl

.1 2 nin{v(6^ kl:q'

^) + v(a- .) + v(a^ .)}

F,K,f

L,)

Krl

= m i r i m i r , { v ( 6 o , - n ) + v ( a , -. ) + v ( a n * ) } , l-t) k't F,K'{ kfg.

k+.R=,3

)min{

m i n i v ( 6 o , - n ) + v ( a , -. ) + v ( a o . ) i J x',1 K,r D,K,l kfg,

k+.q,> 3

n,ir, {e3 - 2 + l9#f

. t]$

,

kl:t:' k+ 1'='3

n i l r, ri r, r. Jl r o) r ktt9" kr.J!>3

. +, r 5 ( k + l ) b | ,--l

2 m i n { m i . r {r g s - z + 1 5 ( i +

k"l-JL>3

+, rL 5- i

- k \ 2'--l

j)--(x+gt-tl

kltit' k+'!'==3 n r j - n{ g B -

3.r

- (2k + ? ' ) 6 + [5

2

Bl * J + t[ 5 ( i

'-r r 5 j |

-

[ 2

,

+ j)

(tt + L) -

2

3 '''

1r.i III

a a

The minimurn Crf the

v(e*. i,l p r L ,

latter

r ' r o ws ; h o wt h a t

+ il -:lJ }

= 93 + 3.

u ( r ,g , r , r )

Frorn (1.1.9),

(1.I.10)

we obtaj n

-L,.J.

d,,

.

5 ,

6rt

=

h,.

. 4 , ' L

L Z | J

:

.

IO4

x

. = 5 Z* 5 4 ,

a . ^ = f l . Lrl

5,

DrJ

d. Z t L

.

=

R." l a r n

,

-

[,]-]gY

t'4

hav 3

t h e n

a . . a yn. t ) . ) 2 g B + 4 .

"

Krl

FrKrX,

foLfows

: L + i = 3

; r n d

v(6^., It

so

+ r)__-_9"1

a . . ,= m - ^ = 6 3 x

S o i f

, eB + 1!1i

)

and (1.1.11)

We also

k + . ? ,= 4

at1-ained when

)- rnirr{ sg - 2 -t 11G-L-!---91

= eB - 2 + [l]i We will

is

term

that

tB,L,,=

T

f

' t 8 , k , 9u. k, , Lo L , 2

*1, n!, = 6 8 , , L , 2 a " . L , ro 2 , 2 * 6

8,2,'r

u2,l u],2 *od 59BJ-4

x 55 - 5408 x 55) *od 593*4 = 68,, L,2$.Lsol q R + a^

q

= 4 6 - - 8 9x l " A B , I , 2 m o d 5 ' ' "

u(6*,r,r) = gF,- 2

so

,g,r,27

o *o,l 593+4 and u(,g,r,r) = 98 + 3'

Now supp'rg5s

u(tB,r,r)=9J+3 and

u ( u * ,j . , j ) ,2 e g - 2 . 1 1 t f j l _ _ - 9 1

From Lemma (2 .2 -5)

wt: irave

68n',i,: = *li

?or i + j ):.

u,!,

u ( ) , k , tb k ' i b t " '

and''S'k''q' = 0

for

k = !'

c^

l.\\r

T.6mhi

v(6.. F+rrlrl

f )

).2)

.) )'rnin{v(:^

" ) + v ( b K. r l . ) + v ( b ^L t J. ) }

DrKrX,

1.J. Kf /..

- . :*m . i t- n [ ** rl -l = t

tr .l '' ,(/ e g , k , . ( t* )' ( o u , i )

* u{bt,j)},

1-JA

\tL

k+9,=3

+ v i b k , . ) * u { b ! .j,) } } {v(e u,k,t)

lK1f L: k+.0>3

)> min {

*i.' kl9.

?'

' 5i - a" --31]

2'

{.rrl + 3 + [!i--:--]t--

,

K+.f =J

m i n { g F_ 2 + t a t + 4 l - " '

+ 11:-_-f 3l:i

.t:!-}:_?t

kl9, k+g>3 ) > m l n{ , , i .

{L -q/ pB + 3 * 1

- 51 - (k 1 [) 1,

5(i + j)

2

k:.19" k+.,Q.= 3 nrin tgg \:19k{-.Q.> 3

The minirnum f the

lettter

2; T+LI -5 ( k

is

tenn

+ &) 2

attained

n o v rs h < i wt h a L

(1.1.1 0) and (I. I.11)

l r f

s g+ - s + t 5 ( i + l )

-el t

- 9] = e ( B+ 1 ) - 6 + I I l t i X f

V(6^.. ' .) = 9$ + 7. p+LtLrZ

= ' n 7 , 2 = 2 8 x 5 '

= Io4 '

b 1

From (1.1.9) ,

, 2 = ^ 7 , 3 = 4 9 x 5 4

b2r2 = mi3

=

'

364 x 54

,4

have

u ( ' B , k , . Q , b k , i l " ! >. ,e1B + 5 * 1 - s - ( i - " " # l S o i f

so

we obtain

'r,l-='nr3,3

We also

+ i ) - ( k + 0 ' )- --5J r' :f rl z

l

= e B+ : + [ l ( ' l i ) WewiII

r5(i +t---

k + 9" = 4

when

v ( 6 D , l*._ , . 1 ) m i n { e B + 3 + 1 ! l : - : r X - e 1 , F+I ,

6- rl

k + 1 , > 3

a n d

i + i = e

L h e n

for

k + q'>3'

34.

, bK. r l b ^ . 1 ) 9 g + r ] .

v(c^

firRtt,

"1

It

tl:rilt

follows

r . 'R*r-1-2 b+L'Ltz

f

t L ',it

Ll ^ ' - a D1 ' ' "a9 , , 2 t Krl gr-"t F,'K,x'

='trJ,r,2 br

,rb2,2

q Rr-g b 2 , r -u I , 2 I n o d 5 " '

* -'g,2,L

f e,. , ^(10192 x 55 - 5095 x 54) ,.nc 598*8 l)trtz

q Rr-q

/r

= 4 : ; 8 6 4x 5 ' " e 8 , 1 , 2 * o d 5 - " ' "

v(e^ . ^) = 98 + 3 J_ 2 F,

v(6...

.

7 0 m o c l, 9 8 + e

s,> 6.."

^ p + L , .! | Z

r

-

^ ) = 9 8 + 7 , = 9 1g + l )

L e m m a( 2 . 2 . 6 )

2.

p+!, L, z

and follovrs k,y ir':duction

o n g rFha fol lowino

rosult

rv-'l-

p(6-) = X^, , = 3'(],r r

u

Proof.

Suppose

-p ( i ;- 1 = * . . l. rrr 0

is

odd

=5

and

n

.O,+1

x 5* mod t

sothestatenLentist-ruefor

p(6^)

= X^ w ,

u

Lemrna (2.2.I)

[161.

a 2 I,

For

L e m r n at 2 . 2 . 1 ' )

llatson

to

was known

t

=

r u - ]3*

x

o=1'

_G _ _ 0 , +-l . 5-' 5'mod

I'rom

*

rve have

V(x^,.)))u*1:i;! 2u+2

ror

i>2-

olrl'

Hence f rom ( -.. f , B) it

follcxvs

xo . J - = r,r

I

that

x' o , i

ui,l

ilr = x . a . . o o d 5 0 + 2 Lrr' 0'1

Now suppcse

c'.

is

= 63 x 5 * *0,,

*od

= 30 x 50+1 *od

50+2

evcrr tulcl

n(Co)

= *u.^

50{-2

= ,n-t

x 5fr mcci :ifl+l"

35. Fr:on Lemma (2.2.I)

vre have

] ; - a + 3 f o r

v ( x . ) ) a * [ 5- i = 4 ] o(ra

(f . I .8)

Hence from

2

it

f ollcv:s

i > 2 .

th"rt

:x 0,r1,1

f

b. ) x 0,a r,r ilr = x - b - - * c c 5 c + 2 rrl-

0r1

= 2g x 5 x x -

A -L,

C X ,f

^0 .. -0,+I

-

Lemma (2.2.7)

fol-lows by induction

We are norv in

a posil-ior-) to pLove (2.1 .4).

Theorem (2.2.-8) . For k

not

divisible

(2.2.gJ

in

6- -0

the

sense thai-

Proof .

the

Su;:'pose

for

5CX of

does not

hold

a = 23 -

odd with

is

an inlegral,

constant

n 2 O

all

modr-rl-o

congruence

0,

exi sts

= k - * 5 2^p ( 5 d n + 6 - ) r , o a S [ 0 cr

^) a"+2

th,. rncinrnnal

iS

-a+2

o .

there

trv 5 srrr:h that

p(5o+2n + 6

where

on

C t> I

every

. mod 5-,-

7a/2]fi

,

24, ancl this

best. possible

for

a higher

power of

L,

say.

5.

From Lemma

(2.2.6) we have

v(6^ F r l r l

,) )' 98 - 2

or (2.2.IO)

From

there

Letnma (2.2.7)

is

qR- ?

X^n , . X^^ - . I zp+Ltr ztr-J.t) follows

that

V(x^.

, )

= C(

k^,

with

(k^D ., zp-L

5) -

it

an Jntegcr

,

zp-L

X^,... . - ; ; z- :l: 7- + r ' I

,^ ^ 11\ \z.z.rL)

X^n.. lF+rrl

x^^ . zD-rt1

X^^ . tF-!, t_ ----i--:.zP-L

_, = K^^ . ZP-l

rzla+!

u r f

J

mod

Ct >

for

I

L.

Therefore

such Lhat

-. ^ _ap-z 5

. flo(l

f

)Q-1

=0mod5-'

x^,.. zE-Itl so putting

i

= I

Y

(2.2.10)

into

-

zP+Itr

"z3-t,t -

5tr-

\

and divicling

- -2 V

n

*21.n:r ,r

I

t

5zt'rt. :

K^n . ztJ- r

^

) )

both

sidcs

" 2z (3' -- Jl , J ,l

^

d

xt?-t - - -a-f: - - ; -r 5z:-

t

t-

I r

5

by

? R * 1-

5""

vre obta.Ln

- -'7g-r

-

X^.. Inocl . l.:- ! 11

,

F t ) "R-l

.

36.

Hence

X^.,.,

= k^^ . x 52 x^^ .

x .^ +t+Zr)

= k

zlt+r,)

l-rom Theorem (1.1.6)

t1-t

it

CL

x 52 x

follows

n )

0

is

0.

W e ,v r i l l

mod ,lla/21

0,l

+l

that = k ^ x 5 2 1 , ( 5 o , r+ 6 ^ ) * o a s [ 7 a / z ] + l 0, cl,

p{5o+2n + 6^,.) cL+z for: all

mod 57P'-I

2b-1.,)

(2.2.g)

novr show that

is

bcst

,

possibJ-e rvher-r

odd.

Suppose

..28+l p(S'p"n

for

n )> A.

all

Then from Theoren

* = *rR_r- *

* 6z6ur)

(1.1.6)

2. 23-1 5tp15'p-tn

/ , 6 1 8 _ r , r n o d 5 "B

*

we have -t /r -(5st 1 1 + x 2 B + t , 2 q E ( " f) r - 2+ " '

-r r;(- s 5t5 ?R+t h ) - ) q D = *x2- 3 ^ +- r , t ; , " f I p t 5 ' ' D * t r +, c 2^g + t/q n)0 -

!

!

\Y/

= *2g*l,r * (*zg*t

,2

\\1/

+ 6 n 2 B * 1 , 1q) +

and

5

5) ? R --. .1 0 (o5) r (aqo * i.z. ' _ t ) qn" = * z ? , _ L ,= f - * * z 3 _ t- s - : : r 2 ) p(5-" ,2

n

n>0

=

"2g-r,r

*

E(q)

!,\.1/

_

It

n ( * r B - t , 2 * 6 * 2 9 - 1 , 1 )q +

Therefore

* z 8 * r , r = k l 3 - r * u ' * r ! - r . ' * o d s 7 B' *zB*t

* 2 6 *x2 ^g - 1 , ) m o a 5 7 B t - 6 * 2 g + r , l f k z B - : .x 5 - ( * r B - a *+ G r) ,2 ,2

and

- 5- *

v^ 2 8=+ 1t

,2

X-

X

^

"28-r

. - ,2

.

-

:

^ n , 1

5zp1-!

-

m

-

-

V

' ! a n

!p-I

n -

n

n

' ' D

5t

o

z D - ) .I z

*2F-],1

xz8+],l J

^

t

v-t

d

-5 - 1 3

-5E-r

and

Therefore

1agtr,;l "?.i:!r

=29:]-,J}it -

-5 2 ( -- L \

- 2 8+1 )

5

-23-I

l

n,nn "5F-t

-zE+I 5

qR-'l

*2g.r,Il. '2.3-l

= *r$-t,2x23+L,r

,l

mod5-"

*

or O^ .. ^ = DtJ-tz

Rrtf

frnm

T,omma (2.1..6)

js

Heircc (2.2.g)

is

cx

5

V(0"

rvr: havc

besi- possible

Now suppose

-eB-1

U mOd

=

. ^) Pt-Ltz

when

is

o

98

-

a

contradictjon.

odd.

a = 23,

even with

2,

say.

Fron Lernma

(2.2.6) we have

v(€^

F r l r J

.) > 9g+ 3

qR+q

> < 2 8 + 2 , i x 2 3 , ) = * 2 3 , i x 2 r . . . v r , j* o u 5 ' '

\z.z.Lz)

Frnm

T . o m m z (\ )4 . - ).

Ther:efore

, 1J )

there

is

it

an ir-rtege,r ' x23+?.,r

\u ' ' ' Lrt

I

)

/

|

-

< l

)

-23.F2

V (. x. -. 0 r 1, .) . =

tirat

follows

f

k^. zp

wit--h X"P '

0rg

;iit

for

0-

(k^o, zp

"

O )'

5) = I

t.

such that

-58+r rnod 5

,

_28

X ^ ^ . =* U m O U ) 2 1 1r )

so putting

i

-

I

in

(2.2.12)

ancl dj-viding both

sides by

528

we obtain

x2g+'2,i

laur ,l;

!'-rs:3-tL^* z t^, , mod 518+3 = J = 52 j ;rt,2-

) xro r - k q " --:r-l- rr., x - "28

s2B

j

rR-r? *od 5''""

38. Hence

2 . r7B+: *zg+z,j= nrB * 5- *zB,j *oo or -= u" 0 * 1 -2 " " c x , m o d-r [ l u / 2 l + z j "'--

^x J - F 2 , j From Theorem (1"1.6)

follorvs that

it

p , 1 5 o + 2 r* 6 o * r ) = k o * 5 2 p ( 5 o n r - 6 o ) r n o a s [ for

al-l

0,

is

n )

O.

(2.2.9)

now show that

We will

is

7d/2]+3

best possibler vrhen

even "

s u p p o s e p ( s 2 3 * 2 , -*r 5 z g * z ) = for all

n2o

Then from Theorem (1.f.6)

r I n)0

* s 2 ( s 2 B n+ 6 r u ) n ' o d s 7 B + 4 ,

4g

we have

* ,t :. 2 8 + 2 0- c2 8 , +, lna = * 2 s 1 * 2 .r r( q 5 ) 6 * x+2v0^- r 2 , 2 q = 1 3 p n+ . : / * LtJ'-' " ue i n t l l ' + . . . (q) e

tq)

= x2g+z,r + Grgrr,2 n 7 xzg+z,r)q n "' and

* 6z,,)q'= *2F,, +xz;,,n.i$# + "' n(r28" ffiF "lo = *28,r *

Grg,z n 7*2g,1)q +'

Therefore

* X^n.^ zp-fzrl

,

= k^^

?

x

5-

X.^

zp

zDt!

-7R+4

mod 5'"

.

* z g + 2 , 2* ' / * 2 g - r - 2 ,=, u * r g* s 2 { * r u , 2* 7 * r g , r ' * o d 5 7 8 * 4 and

-* -2 - -76+4 moo) X.^,. . = R.n x 5 X.. | z 1 , "z 26 26-12,2

x28,L

x2?,+2, 1 -.* _

-

-23+Z

f

v

,

. -53+?.

m - ^ ' )

zb

^ a

-zl5

J

39. and

xz3+2,2

x2.3,2 . 5- 5 ? , + 2 = *rB --Jf mocl

sr{;, Therefore

Xro,r

'X - , o

2 6 + 2 t 2 -ef il\,L -Fn= t

r

X.n

Xr/l

z p r .r

/ . , T, rL ,

;fr-

r

;,s;z

moo)- 5 3 + 2

- -93t.+xz}+2,2 x2B,r = *2g,2 x23-v2,L rnod5-or

c.-^=

omocrs(]+a

D t ! t Z

Frrr

frnm

r . a m m r (\ L2 .. a2..v6t )

we have

V(e ^

.

^)

= 98 + 3,

a contradrction.

D t l - r z

Hence (2.2.tr) the proof

of

is

best

possible

Thcorcm (2.2.8) .

vrhen

cl,

is

even.

This

completes

40.

2.3.

of the calcul.ations

sect.iorr we provide sorre details

In this

f o r ( 2. L . 2 ) a n d ( 2. 1 . 3 ) .

'I'neorero

\ 1,- .J. I) .

p(5o+2no 6o*r) = ko x 52 p(5on + 6o) nod 5l'ta/z)

(2.3.2) L ^ l 1 -

,,uruD

. , l r L

'-

1 -

wrLr,

^,

-,,-J ^^1,, artu vrrry

x

k3

{2.2.L3)

l nr^ri

*o,l

nn

If Iz f:

are defined

C l, l

arrf

(2.3.4)

and

3339,

== 240839,

i-t

k^ :, f1:++] i' . +l \ 5'"'" I 5*

where the

*o*2,1

=

k4

foliov,'s

=

2193964

that-

ar-rd kr

(2.3.2)

= 256t00:2 14'

Lro}ds if

;€ rr

(2.3.3)

fr.l

k2

Frorn (2.2 .lL)

Proof .

.|-ha

89 ,

+z

in

,rou ,[lu7zl+z

.rAn.(rr1/trn^-.

*^.1

t

*ouutstlz)+r

)

(1.1.8)

for

.

Hetrce we nced onll'

r ( o,(

s.

calcul-arr':

we have obtained

q20

= (5, 0, 0, ...), = ( 6 3 x 5 :,2 5 2 x 5 5, 6 3 x 5 7 , 6 * 5 1 0, 5 r 2 ' , o , 0 , . . . ) , = ( 1 3 5 3 t 3 3 9 x 5 31, 8 8 5 0 2 6 2 21 x 5 6, 7 2 0 1 3 3 3 x 5 9 ,7 3 0 7 6 0 8 x 5 r 0 , 5, o, o, 72766x5r3,3l-3x5f

ra

5 5 7 x 5 4 , ] 0 1 5 8 t 5 6 2 g x 5 7, = (5549,1083 -

Is

-

8 7 ' 7 2 2 3 2 x 5 9 ,L 7 9 7 8 4 x 5 1 2 ,

3 3 3 g x 5 1 4 , 00, , . . . ) ,

2, ( 1 0 1 9 2 3 8 7 t 7 1 x 5,5 1 4 3 1 6 2 4 9 3 > (, 5 88 6 ' 1 1 3 7 x 5 1 11, 4 9 9 8 7 ' r 5 f 4 9 x 5 1 5,

Ia

. . ,) ,

r o - 7 x 5 r ' 7o , ,o , ...) ,

= ( 1 5 5 8 0 6 0 1 4 8 x 5, 6 2 9 4 0 6 8 9 2 . x 5 92,7 0 1 4 8 x 5 1 1 ,1 4 4 0 1 x 5 1 4 , 1 7 r x 5 1 6 ,o , o , . . . ) ,

and

fz

. , ( 2 6 3 5 6 3 9 6 g x 5, 76 1 g 2 8 7 7 x 5 1 0 1 , 7 3 1 . 8 x 5 1 33, 4 3 x 5 1 4 , I I l x 5 1 7 , 3 " 5 1 9 ,o , o , . . . ) .

t1

Frorn (2.3.3)

r

4

A11 these in

and (1,.3.4) we obt_aj-r] k, = 89, k^ = 3339, r z

about

ctrl-culatior,s

tsrom this

\ . , ? € r cd o n e o n t - h e c y b e r

2 s:econdsexecuti-cli

The author

has in

his

t--inre, and these

possession

t;rbre we have been able

3 and sma1l values

if

5

of

n.

a tab-l-e of to

verify

o- ( x (' - ( l ,

,[s,-t./z)+t

r71. compute:: at

u.li.s.l.l

ha.re been checke3

f

chapt-er we o b t a i n

and

7.

t rn

analogous results

llatson

[ 16]

i s the recip.roc.rl

to

(2.i.2)

obtained

t-he fotlovrirrg

rnodulo

7a

oi

and result.

24 then

fcr

n)- O,

( *oa z[ a/z]+2) ,

I -= l1

p( tr *) tl$+- !L

(3.1.1)

p (70n + ).o)

II 2 4 p ( I t

For

_

and

(3.1.2)

p(aen + 4!) u/ z) -vt, ( m o o ,l t a

. - r u)

p(343n+ 243)

even.

49

(3.1.1)

A

oc1d,

i

c

aa'.i

rzr'l

anf

t.a

5 x 7 p(7n + 5)

mocl 73

and p ( 2 4 0 1 n + 2 3 0 1 )= 4 1 x 7 p ( 4 9 n + 4 1 ) m o d 7 l Using are

methods cotnpletely

able

to

obtain

(3.1.3)

the

aneLlogous to followinq

those

of

new consruence

the

prer"'ious chapter

v,,e

relations

p ( 1 6 8 0 7 n+ 1 1 9 0 5 ) = 4 3 9 x 7 p ( 3 4 3 n + ' 2 4 3 ) m o c l7 7 , P(117649n + 712741) = 524I x 7 p ( 2 4 0 l n r . 2 3 C l ) m o d 7 9 , p ( 8 2 3 5 4 3 n+ 5 8 3 3 4 3 ) = 3 7 4 9 9 5x 7 p ( 1 6 8 o 7 n+ 1 1 9 0 5 ) m o d 7 f I . ' p ( 5 7 6 4 8 0 1 n* 5 5 2 4 6 ; 0 1 = ) 1 1 9 8 5 3 8x 7 p ( U J 6 4 9 n + i - l 2 ' 7 4 ' 7 ), " o d 7 1 3 , p ( 4 0 3 5 3 6 0 7 n+ 2 8 5 8 3 8 0 5 ) = 2 4 3 3 2 O I B Ox ' l p ( 8 2 3 5 4 3 n+ 5 s 3 3 4 3 ) o . o d 7 l L 5 't

and In

p ( 2 8 2 4 1 5 2 4 9 n+ 2 7 0 1 0 5 4 4 7 = ) 1 6 5 5 6 9 6 4 2 5x 7 F ( 5 ? t , 4 8 u 1 n+ 5 5 2 4 € : 0 L m ) od ?

fact

there

The main result

exist of

similar thrs

congruence rel.ations

chalgsa

is

the

fo1.l-cwing.

for

aJ i

highcr

powers cf

7

?

cx> I

rf then

there

that

for

exisLs

all

is

constant

the

reciprocal

L^

not

modulo

divisible

by

,cx

of

7

such

g

p ( 7 o + 1 2 *r . t r o * r ) - g o 7 p ( 7 o , . ,* t r o ) * o d 7 2 0 + 1 is

best

In

section

possj-b1e.

3.3

, -0

P(7-'n + I^) Ci

which

tro

an :Lntegral

n )

(3.1.4) and this

and it

enable

us to

we give

the

1'l

g,^. C[

(3.1.2) and (3.1.3).

functions

(1 ( a < I0)

mod 7-'

calculat,e

generating

for

I (

cr (

for

, g

thus

obtainino

2tt,

44.

3.2

As in

2.2 we '3 )

-ir. "'*"

{ t

-i.

klt' k+.Q,==3 .

r en

ml-rl

t

5l)

-

J

+

r 7(k + l.)- f2l l-----'-l

+

tl -' 1 ( i *

j)

-

(k +' t)- I'r'.1 -'l :j

klt', k+r,>3 The minimum of

the

latter

is

term

attained

u(r',i,l>nin{:;g - r+17(ia-i)---8i .,

)min{sg + r + t1!Lrj4l--:-J9

We will

-

_ f c, .pe t+ rr - _ = r - t[7 ( i

+ j)

now show that

,(u*,rrr)

(1.1.19) and (1.f.20)

trrl

=

uz,L =

k + 9" = 4.

when

So

58+}+llJ-lji)

sg + 1* tlJ:-r}-Zt

16I = 53 -

2-

From (1.].18),

we obtain

^4,2:

82 x 7'

* 8 , 3 = 3 5 2x 7 '

- el

dl

,2

= ^4,3 = r'6 * 73 '

u 2 , 2 , =* 8 , 4 = 4 8 7 5 8 x ' / 2

i

I

.

47.

w e a r s o h a v e , ( 6 8 , u . , f ,u k , i u Q . r - s B + r + l Z J i - 1 - * l - L : 2 1 r o . ,j) S o i f

k + g > 3

e L n di + i = 3

^ a,

v(6^

Krl

Brlirx,

It

foll-ows

x + . Q) ,: .

t h e n

a^ .) )-5? + q. L,)

that

t B , r , 2 = - e p ' , , 2 ,=L'

ol,

- u g , k , .af k, , r a g , , 2

nlr

- D

= 68.r,2 ar

uz,2* 6

,r

s,2,L

r 6 . , ^ ( 3 9 9 8 1 5 6x 1 3 -

^

u2,r tl,2 *od 7f,p-fr

6 1 9 5 2t l 4 ) * o c 7 5 6 * 3

D r ! r z

= 3 5 6 4 , 4 9 2* 7 3 6 ^ , ^ m o d ' 5 8 + 3 F r L r l

= 5g -

v(6a . .) D,rtz

I

so

Now suppose

5R+?

t)n . ^ 7 O mod 7-'p--' brLr2 V(e^

t
rl-,2'

^) ) = 5 $ + 2

and

v(r*,

r-sB + I . tJJi_3--])_-_J91

r,jr

From (3.2.3)

wc have

e. ,_ n = 0

for

6^ b+r,1'l

k = !,

t rir

I [lr

e^ ^ F,K,r

i + i ) 3.

for

b. bn L,) K,]

ancl

so by Lemma (I.5.2)

F t K t L

v(6^ .

.) ) min {v(e^

F+rrr-rl

, , ) + v ( b . . ) + v ( b v" " t ,) ) }

b,Krf

klg_

= min { min 13 1.J0

The minimum of

tire latter

t.erm is

obtained

vrhen

so

= 5(g + r)_ 3 + 1i(!_fil__t? 1 .

= 58 + z + [ry]

,(So*.., , ,) = 58 + a.

now show that

From (1.I.1g),

(1.1.19) and (1.1.20) we obrain = *5r2 = 190'

'r,,

= * 5 r 3 = L 2 6 5 x 7 " 2'

br,r = *9,3 = 255'

br,,

= * 9 , 4 = 1 7 1 - 1 8x 1 2'

btr,

Wea]sohave,(.B,k,l,bk,ibL,,125B+4+'.t#fork+l,):. S o i f

k + L ) :

a n d

i + 1 = 3

It

follows

t h e n

b " . ) ) 5 8 + 6 .

V ( : f ^ ,n b . F r K r {

K r }

L r J

that

= - 68*, = 6B*t, r,2 .k I>-r ,2,r ='g,r,,, of ='8,r,,

= 58 + 2

stl

0B*t,

bk bL,2 ^ - l - ut g ,x,t ,r "

L>L

* e g , 2 , r b 2 , 1b f

,rb2,2

,2

*od '53+5

( 3 2 5 2 4 2 0x 7 2 - 3 2 2 5 1 5^ t 2 ) * o a 7 5 8 + 5

= 2 9 2 9 8 4 5* u(tg,r,r)

tl

5 8+ 4 + [ l ( : t + i ) - 1 3 r ]

v ( 6 g + r , i , j )) m i n { 5 8+ z - t l a # ,

We will

k + . Q ,= 4 .

,l

ug,1,2 *od '58+5 (Rr(

"

0 mod7-''' L,2 7

' i _ 5 8 + 4 =

,t \ r- Q/r .A' r v

Lemma (3.2.4)

fol-Ic;ws by induction

The fotlowing

rcs.rlt

'

'

and

t

L

l

on

r , s 4 sk n o w n t o i , l a t s o n

1 L

)t

B l-16].

5 ( B + f ) _ 1 .

dq

L e n m a( 3 . 2 . 5 ) .

For

B2t,

p(tr2*-r) = *28-1,1 = 5B-l * 78 mod 78.+1 and

p(trre) =, Xre ., = 58 * 7B+t mad7B+2 4tst,

L

Drnaf

='

p(It)

*l_

n ] = 5" x J'

= i

,I

? . m o d ,l ' .

Now suppose = *2fi-:-,r_ = ,B-'

p(l2*-1) From (1.1.17)

we have =

X^o . tDtL Soif

f

7 ur,r*12

=

g 2 x 7 2 + Z 5 Z x 7 3

then

:)

u2,r

- - 2o c t -l 3.

m

f r o m T , o m m a( t

5.3)

we have

u ( * z g - r , 2i )B + r u i - r > ' z It

follows

a. r.I

=

= B )

x^^ 26-I,l

I iir

then

B=1

*2,r

If

x 78 mod ,3+r.

3+z

fcr

i>2

that P ( )z D, ^ " =1 X ^ ^ - r x ^ ^ . . lDt!

tl-Lr-L

a..mod7B+2 -Lrl.

= 82 x J x^n ., . mod 7B+2 zP-L, !

_ +r = _3B .^. _t B m

o

^ ( _ g1 + 2 l

.

Now suppose P().^^1 = X^^ zlt 26r! From Lemma (1.5.3)

= 5

B

R+i x 7'"'moci

R+2 7''

wc have

u , * r * . ,>i (| B + I ) . t f l

)g+:

ror i2z

Therefore

( 1 . l. ..17) it

from

f ollorys; Lhat

X^n.. , z p + r ,r

=

I ilr

*^^ b, z 3t L f , I

J

x^.

'

Qt)

l'.

zDrL

. mod 7''

'

l.rI

R +-) I90 X^^ . mod 7"

=

l Qr I

_ Lemma ( 3 .2 .5) We are in

fol_lr:ws by irrduction

a positir:n

[o,

I^ CX

possible power of Proof.

s B- I

F a

(3'2'8)

F r o m l , e n u n a( 3 . 2 . 5 ) i i : f o ] l o w s t h a L Therefore there i.: :r.rinl-naor Xlrlrr

\J'u"'

r


sg+2 K l a P T L ' )

or X^n.^ . zp-tttt

\5.Z.fU)

X^^ , tltt)

= $ + 1-

,)

zp rL

( 9 "^ " , 1 ) zp

I

\ r ' ! ' r< r /

t

is

T'hereforc there

X^.,.^ zb+zr)

-n ^ . -f,p+z mOO / .

"u /r n" 2 6 . ., r,, . , )

an integcr

such Lh.at

x2B+2,1 t

X^^ . tDtL

f o l l -_ u- r-! ^> !LLr r ^d L!

F r o m L e m m a( 3 . 2 . 5 ) i t V(x.n

=

t

l

t

-

-B+: t

x^ 2ft,f

v

-- ' 2^3

*2fi,r h

n

d

-b+l

t

omcdTB-1

. -3li-] t

= B + 2 L.n

zp

wit-h

and

53. q^

nrrtfinft

ri

--

r]

(j \ J . a .2r \ / l/n )

in

q i_r _. l-o* q jhJr Lr l---] -r - _

r rA - iVr za iu; iant nr l j c: ,rnt ;l u U

.l -- rJr z

"8+l

we obtai-n

xzl+2,, f*

= '7;,1'-t- *rs,j *ou 148+L = tra * ,4.! zD

*,,. . mod't|e+L tl1

=- L^n x 7 x^

rric4 ,43+1

t )

TD*t

Herrce

X^^ . ^

zL+ztJ

Z f" i t )

tP

_

or

= [o

x0+2'j

From Theorem(l.l.16)

we wir]

it

t 7 *o,j

*ou '2a+L

follows thaL

now shcw that

(3.2.7)

is

best possible

v;hen

o,

is

Suppose

p 0 2 ' B + 2 n* t r 2 6 * z ) = n r ' x 7 p Q 2 B , ' ,* t r z * ) * o o z 4 B r 2 , for

alt

n )

O .

T'hen from Theorem (l-.1.16)

p t 7 2 8 * 2*n \' r2g9 *+ z2 '-) q n= =" 2*82+8r ,+ r , tr i g ! , n)O u (q) I

= "28,f

*

we have

+ * '2z 8f+ 2 , , ^= E ( q n7 ) B o (O)

( * r g , 2 + 5 * 2 g , 1 ) Q*

Therefore *2g*2,r

__ ^ * :: Lrg''

*zg+2,2*5*2g+2,L =

"rB, 1

A3+2 m o c i7 |

nig*l(xz',2*5

* r B , r )* o a 2 4 8 ' + 2

even.

C.A

and

,ire*,

L'.re *

!

-t -

J *.o

- P

mod ,48't'?'

L l ) r z

"

I

" 2 8 + 2 , r -- , * i Z s , t - ^ o , 3 8 -zB ,B+z ,B+r and

ltzs:lt ,3+z

r he re fo re

= - . 'g* -28

*z?,,2

*^,0 ,38

,g+r

" z:g!34 1Pt 7g+2 ;$r-

- *r 3,2 = /+r.

xz.g+2 = "28,1

xz3

*29+ 2,2

*oa 733 ,

x2$+2,r *od 75Bl-3

,2

c = o m n A " 5 3 + 3 r D t r t Z e n

But

from Lemma (3.2.4)

we have

V(S. .

^) = 5$ + 2,

D r L t Z

Hence (3-2-1) +ehr ar u

nv..^€ 1,r.vur

^4 vr

is

best. possibre

mL^ rrreolCrTr

( 3.2.6)

.

when

cx

is

even.

a contrad_i-ction. This

conrpreues

qq

3.3-r_

In this

s e c t i o n we Provide

sone d e L a i l s

of

tire calcul;rtions

for

( 3 .I . 2 ) a n d ( 3 .r . 3 ) .

T h e o r e m( 3 . 3 . 1 ) - . (3.3.2)

9._ = "qt

holds with

n h l t t

0

-02

!

) rr -t-1

p(70n +

moo /

439, L4 =

1". = 374995, [f

9,, = 24332O1BO and

Proof.

g.^xj

p ( 7 0 + 2 . ,+ "cL+2' ) )

r^ = 1b))oYo4t3. U n

From (3.2.9)

,

b

t

= 1198!,3f],

a

(3.2.11)

and

it

.

fol]ows

(3.3.2)

that

holcls

if

and

t +

xcr+z,t I ["o,] l-i

-'9,I I rqr*l 7 '

L ^ ,= *

(3.3:3)

u

) l t '

where the xu.+2 ,t

*o, and ?nv+'l

*o,I *od 7-*'-

i

? n - L l r " * , ^ I i _ - - - i - 1f .

mod

)

ar,3 d e f i n e d

in (I.I.17). r 0 r r 3 r f ,* f ' t;J+t ) +1 '" 7 '

*o,1 *''d for

t (

cl (

g .

I

Hence we need orrly calculate: or

xa+2,I

We have obtained

the

and folJ,owing

congruenc"= *od 717 (3.3.4)

1

= ( . 7 ,7 2 , 0 ,

lz

= ( 2 5 4 6 x 7 2'

O ,

. . . )

t

4}g34xj4, r4r8989x75, 24g}Boox77, 23g4438x79, g 6 3 3 x 1 r 2 , 4 4 x 7 r 5 ,o , o , . . . ) , - 2525gx7rr,

f:

= G73425r)Is07l2x72 , 834L8g43353x74 , 974544L7x76, r2g3!612x78, 26.4230.2x-t9 , 3 6 0 0 0 x 7 1, 1 2 1 5 3 x 7 1 3 3, 9 x 7 1 5 , 5 x 7 1 6 , 0 , 0 , . . . ) ,

Ia

= (192116(3697gLx73 9 ,8 0 6 3 6 7 0 02 x 7 5, 1 5 8 2 3 8 9 1 47 x 7 6, 3 8 6 1 8 5 9 6 x 7 8 , 5 g g 6 5 0 x 7 1 0 , g 4 2 } x 7 r 2, r 8 z 5 x j l . 3, o , 0 ,

IS

, 3 1 9 3 4 17 r g x 7 5, = ( 4 1 2 0 4 7 2 ' 7 3 s 4 o x 7 35

.. .) ,

2:*262g4ogx773 , 55r650x79,

334569x710,26'76xj}2, 202x7r4, 6x71(',0,0, ., ") fo

= ( 7 6 6 3 7 0 5 8 t 3 5 x t 4 , rg72Ba2rggx76, 2o2g086r4x77, 98'o2r2x19 4 8 1 4 r - x 7 1, 1 2 5 4 x 1 r 3 , 2 7 o x 7 r 4 , 0 , o , . . . ) ,

56. = ( 7 0 2 8 3 3 8 € 1 5 8 0," 7Lng 2 4 4 j . 9 6 : * 7 u 7 1 5 2 2 6 6 * . 7 8 , , nrrrrn*.rr}

ll

1 0 2 6 6 r . x 7 f617, 5 x 7 1 3=, J * J u r ,o , o , = ( 7 3 8 1 3 8 6 9 r r ' 7 r r 7n5z,o ) g t 9 o * 7 7

Ie

..'.r,

, r 8 9 4 4 2 6 x 7 8 ,2 3 r 6 4 s x . L a , 2 g 7 x 7 r 2 , 3 r * 7 r 4 , 4 8 * 7 1 5o, , o , . . . ) ,

.

( I o ( ) 6 9 2 4 4 g 1 : - x 7359, 0 8 9 5 5 T x J J3 5 0 O l o B x 7 9 , Ig = , r o n n * r 1 1, ,

3;Lox1l2,

3 1 6 * 7 1 4 , 3 * 7 L o6 ,, o , . . . ) , o"' and

f,-o =

(1213638485>''76, 13481638x78, 549"772Gx19 , qgzrt*'lrr , 729'x-7r3 , 36x715,2x7r6 o, o, ...),

From (3.3.3)

arrd (3.3.4)

9"n = 5241, L5 = 3't4gg5, .Q,U = 1198538, Ll A11 these in

about

carculations

d = I,

2

=

time,

val-ues of

n.

and

computer at

. Q . ,= 1 6 5 5 6 9 6 4 2 5 .

u.N.s.r.I .

and these have been checked.

tab1e we have been able

and small

g,, = 47, g,, = 439,

2SZZ2018O

wer done on the Cyber r7r

4 seconds execution

from our partition

[]_ = 5,

we obtain

to verify

(3.3.2)

for

Arso,

-CfiNiER ]Y C0'iGRU:liCS Ft;R 4 .I

rNrRopug:llgl! Suppos;e

Chapter

6o

is

l- wc: staterl

the

Hunt had found for rn

this

where

cha;rter

the

reciprocal

Ramanujan-type

the generating

we generalise

p - -J< ,_(n)

50

identi-uies

functibns

their

of that

24.

In

Hirschhorrr

and

p(S,xn + 6,,)qt. I n>0

methods to

the

p , (n) ,

functions

ir; clefined by

t

'

n

(nl n r._L\rrl

) r!

--

.t Y

n

1

(l \a

-

-

n " )' 9

,

n )1

= p(n) .

-P - ,r ( n )

-

ll Il

n::0 so that

modulo

we obtain

Ramanujan-type

idenlities

.t-he

fcr

' fL"u- l^l +u L l o n S

yc sn rnrosrr:ql _ L if , r r V n n

r

(n+(s0-6 )n /5d

r

d

P ,.tn) q

r! n=o0K mod 50 and by finding coefficients below,

in

due to

discussion notation Hunt,

a l-owe:r bound

of in

these

Atkin

on the

identities for

t'21

Atkirr's

proof.

of

5

we are abre to p -

the case

unfortunaLely

,3aflsr to be consistent

and earlier

power

with

the

that

divides

the

Drove t.he resrrl;,

5.

We give

stated

a detailed

we have changed Atkin's notation

of

Iiirschhorn

and

chal>ters.

Theorem (4.1.1) . Let Then if

k > 0

24rn f k n,od p0

( 4 . 1 .. 2 ) where the

p

and

-p- K, - ( m )

e = c(k)

residue

of

= o

= O(l.og k) k

be one of

i:he primes

2,

3, 5,

7 or

13.

we have mod pBd/2

and

+ e

t

R = E(k,I;)

:nodulo 24 accor:Crng to

the

depending on

fofj.ov;j.nq i:able:

ll

and

hJ l

l

tJ l

P

{

L

l N

\J l (

tr3 l r

l '

N

rO J

ll H

P O

P

ts UJ

P F

H (n

o\ ts !

ts \o N O

N) P

NJ (]J

Atkin but

has only

primes ' of

has pr:oved this sketched

The pro.f

briefly

of

Theorem (r.l-.6)

Tl-reorem in the

the

basic

fo110wing

rvhich is

deLail

for

the

fonnulae

is

requrred

completery

the main result

of

case

n - R t ' - J

for

fehr |, v:

analogous

Hrrschtrorn

a{-l-.^v

]-n LV

and

{-}. Utq

LIrr'^+ f T U J T L

urlcl

+ L

I 5

paper.

T h e o r e r n( 4 . 1 . 3 ) .

For q, )

1, a

r o

t^ \cr.L.zrJ

I' L h

P - O( n ) q

A

l \ 1 L = L

( n + ( s c - 6 o ) k )/ 5 d

x

i e t J

t

h ] - r

"-

F 1--\

h

i

I *v '

f,

o

d

l

L

Y

-.]-]_-

x (q)

ht

+K

t

where

E(I

o..>L,

( 4. l . s )

where

0

even,

h l

E (o") "-

i d

i>l

Here

odd,

-fr

mod

and for

q h f

\

"

"

q

' * i , j ') i , j>1.

j

= *6i,

M =(i*i,j)i,1-,

rows o f

M

are

aAA

x -0,

/:r

ti,

,

and

s

=

(b.' L

i*i is

,

1-

"irj

defined

. ) .L

t J

t

i > lL

are

defined

by

J _

'"6i+kri+i

as before.

That is,

the

first

60.

5

U

2x5

tr

(4.r.7)

9

n

J ^:)

0

f,

0

U

0

U

U

U

a

zt

4x5

_l

and for

>

i

4x5-

zz^a

A

m . - = 0

q

x

0

and for

1 - r1

(4.1.8)

-5

8x5

0

f

J 2 2,

2 5 m .1 j , + 2 5 m -. j . r- 1, l--L r-z tJ-!

l r J

+ l _ 5 'm i - 3.,

+ 5 m " " ' : L _ 4, j _ I

j-l

.l-

m +1 - \ J r

ft to

is

clear

Hirschhorn

witi the

that

for

the

k = 1

case

and l:lunL's main resul.L

k=1.

t h e vectors

same as those

x

(Theoren

Theorem (4.1.3) ( 1 .l .6) ) .

and the matrices

-cx

:Ln Hirschhorn

and Hunt's

a -J

| 1

reduces

Note f irst

A, B

and

M

are

paper.

Now t ', ,.0

( n + aq O - A v

I

p,(n)

I

N I

J

/

n

n n + I) p (5-n+6^) q-n>O

=

Y

h-n av

mod 5* andwith

k=1

(4 : .f .4)

reduces co t

r

x

)

ilr r

l - l

o ( ,1

q - *

h 1 -

I

E(q )

odd,

h l

l, (qJ

^

/ P ( 5 n + 0A ' ) cr r n20 5

r

)

L

i>l

which

is

( 1 . r . 7 .)

d

x

i

e' 4

j

o

1 - l

"

\y

h l

/ h 1 +

r "/ ^ l " - ' -

|

that

61.

4.2.

We now turn

denote the

the

exact

power of

F = 5

case

to a discussion 5

we will_

mln m:

dividing

=

min n,

z4m=K

n=O

r

d

nloo 5

From

(4 .1 .4)

(4.2.I)

rioo

.

Theorem (4.1.f) n .

find. a lower

v ( -t '- ,K_ ( m ) )

f

of

.

Let

(4.) .2)

To estabrish

bound for

the

V(n) for

follorving

V (' 'p- .k( 'n ) )

k

_G 5

we ha."ze

min m: 24mlk

v ( -p-_k ( m )) ) m i n i>l

v(x .) 0' i'

- - r y

mod 5*

n = 6

Suppose

0,

k mo,l 5o

for

then

( n + ( 5 c x- r S o ) k/)s d > t so that

for

small-

i

6

C

n ,

For insi-ance,

of the

the

= 0"

xa , I,I vector

expansion

interested

in

k = 6

and

= 4;; n = 6 t< = 4 nod 5 and I 0 ,

= 6. X

(n + (5o - 6^)k) y5d )- 14 + (5-4)6) /5 so that

g = t

if

f

e ,

then

k

for arl

= 0

*^,

certain

We shall This

Ia of

ej-ther

now cal-culate

depends on the

side

of

n i 6^.]< mod 50 cx

sr:ppose

largest- possible

m

the

in

integer a9

first

n = k6^ - 5*m 0,

(4.L.4). n

- 50mlo

So

k 6a- - 5 q m ) 0 kd

ct -.1

v

and

k dd - s 0 ( m + 1 )( o k6

^ 1

d a

for

q

,

in

So we are which

then we want to find

such that kd

enfr',2

non-zero

power of

smallest

Lhe equality

Lhe s;ma.llest non-negative

= 2

the

ko 0 r

I

' - 0 '

It

follorvs that

., .

(n+ (lr0- J - ) k ) / s *

,. I

u

P-Jl, - ( n ) q

nl6

,

!

t
l

where

r! \. y/ ^ - \

-

, /

even , l + r . ,0

ry

E (q)

( 4 . 1 . 8 ) a n c tr h e

d

CT

are defined

\ 4 . 2 . , 1) .

Now ( 4 . 2 . L ) l S e c o m e s (4.2.7)

r .\ v lp_u (m) J

mrn m:

( . , \4 mln lv(x^ i_;

. \ )J

i > l

Z4m=R - _ G moct 5

Before

we can calcr:rate

(4.2.1)

a lower

we need some Lemmata.

Hirschhorn

bound. for

the

The following

right-hand is

side

Lemma (4.f)

of of

a n d H u n 1 r - , t sp a p e r .

L e m m a( 4 . 2 . 8 ) .

v(m.-j >lL(sj-i-1)l

,

L t f

where the

R;

are defined by (4.L.7)

.

tr

and (4.1.8).

-J

As an immediate

col'rse,quence we have

the

following

Lernma.

Lemma(4.2.9)

v ( a . ,. ) ) [ + r s j - i - t ) ] , rr]

v(bi,

l">

t ! ( s j - k - i - 1 )l

Proof. B y ( 4 . f . 6 ) a n d l . e m m a( 4 . 2 . 8 ) ,

'

v(a. .) arl

v(ni.. .,.) 2 [!(sti+jl

v ( b . , )

V(nr^,_rr. >ll(Sti+i1 ir-i)

h l

rt)

The following

is

f

4 r

Le:mma( 2 . 11) of

- 6i - 1)] -

l{irschhorn

(6i+k)

I

t

t - ,

t 1()l-]-t)j

. . 1

I

r)l = [ttsj-k-i-l)]

and Hunt's

paper .

in

o+.

L e n u n a( 4 . 2 . I 0 ) .

A - 0 '

t-ho

ron

moCul-o 5 0

i nracr'l

of

^-+.i,-t.:

24,

^^

li., 1 and for

d2

L,

tI

odd,

I

A ' CI1-1

Lemma (4 .2. 11) .

4 x 5 0 * 6 0 ,

t

3 x 5 0 * 6 - o ,

Fcrr a2t,

d I

0 i

t ; - J

,

)

v g u ,

cx+l

t where

the

d +.k 0-t 5

cx even,

J

are definn e d b y ( 4 . 2 . 3 )

d rY

rk6

(4.2.I2)

d

rl

Now suppose

- ,K - 1l - -

'1

0

] | -cx .Jt -

0,

is

\

1," t-

(}+I

Now suppose

c[

9,.,

, .

u+-L

r

f"

l'

'. -_ lt ;i

0

, K a

.v+l

=

I

,

t

)

t " a I _

l

5

l

l

0.

.J

I l

/

1

,

l

=

l

I

t l t b

-

I

+

I

1 - cx+l

f,

( )5 - o

l

J

=

t-

5 -

A

()

I

, ]

d

l .

l l

we na' e (5

n+l

I '|

i I |

3 )

t

/1

t

l

=

5 )k c)

3 xx 5

(

l f , L

I

a (f, f,

L 4X

f,

0

l J

€ I , ll ; r

q

L -] J L

av+]

;t o 1l

I

t

_0+

r\

l

I '

I 1l

.

t

'|

^ ,

( t_-

,

From Lemma \ / + . t . L v )

l t

tt. -

d

() I . ( 1l r 7-

I

J

=

l*

A

r - l \

t'

I I

!'l

even. r

L'

_5

|

^*

l l; tl _ ,- ( t

is

1

5

-Ct

L:

1_

ryrt

) - o

=

- l a

_ _ c | =x | l_

From r Leruna ( 4 . 2 . I O ) w e h a v e

odd.

r, -0*1 d

rI - k n

= k +

I

-

a

ld +k I l n I rl i t--:- 5 t

J

l

I

I

)j

we will

noh' in'!,/estigate

v(x.,, . ,. ) u, 1l-oct

for the first

few values of

g

From Lerruna ,4 . .1. B) we have

5(i+d.)

- '(mk,i+dr) >'t----!-

u(*r,i+dr)

Suppose

k=l5r*sr

where

d : . : t: g + - ; =

5 k I * k

- k -

I

5i

+ 5d-- k -

= t--_-_-rt---;

2---l

1

thenfrom (4.2.12)wehave

0{s(5,

=,*;t, =tA:;*

=r,

5 r - ( 5 r + s ) = - s

and

v(*r,i*ur) - v(\,i+ar) > t!#r so that

this;

modulo 5.

lower

bound depends only

Puttin,g

= I

i

on

i

2 mod 5,

sidr:

drrcl O,

of

this

( t

(4.2.13)

inequatity k = 3, 4t

when

resiclue

of

k

we obtain

Y ( x. r 1r r,r u*l^ ) u ' * u * {' 0 , t + l Z The right-ha.nd

and the

is

} . equal

to

I,

or 5 mod 5.

i r

9 . = - L or

i f

L = 3 ,4or

when

So if

k = I

or

we define tr,

9

?

lllvu

mnzl

J

v

rl.vv

J,

,

e(!,) = .i I

I o then we have (4.2.14)

1,r(x1 r.r_a)>0(k),

or

(4.2.15)

ul*. t'

Later

we will-

k l s m o d 5

r k + 4 r l > 0 0 . ,1 K,l. -;--l I

=fTof

5

in',res;ti,qate a n d

since l+dr=1.+f+l

when we have

0 . 1 s ( 5

t h e n i f

equality

in

(4.2.L5)

s = 1 o r 2 v , ' e h a v e

.

Supoose

615.

v(*l,i+d,) > Jlf----:-:11 t s ,= 3,

and if

= 1 + f 1---r-1jf

z

4 or

5

2

> e(k) * 1i.'_:_!1 '

we have

Hence (4.2.l-6)

61

v (x,1 > 0 (k) * max , t l-f; {o , ir.ar)

Now from (4"1.5)

and (4.2.4)

x 2 , 3 . + d .=,

we have

ria

*r-,i ti,)*d.2 =

*l,r-*d,.i+dr,j+d2 rl,

v("2,j*ur) t Tli

{ u ( " r , i * a r ) * . 2 [ . 1 + d 1 ,*ju r J ]

) min(Ott) + *.*{ 0,f 11-;il i>1 + *-*{0, The minimurn is increases by at

the

att;ii-ned

at

i

secorrd i:erm by at

most 1) ,

) )

(since

least

2,

+ 5d, - dr -

[ !(5j-i

i-ncreasing

i

by

and decreases

the

lasi:

r)] ] 1 t.erm

o t x ) + * u * { 0 , [ ] : ( 5- j + 5 d 1 - d , - 2 - ") ] ]

: r -i ) .t , < t 2

it

= 1

}

so theit

'u(x2

d, = 5r + s

Suppose

}

2

where

O(

s (

a

I

then from Lenma (4.2.11)

foll-ows that dr

5u, - dI = 5[tr]

da = 5r - (5r+s) = - s

and

V ( * , - . _, . . ) > - 0 ( k ) + * . * { 0 , I r : ( s j * s _ z ) ] ] ztJ + ra, so that

this

modulo 5.

lower I,utting

.bound depends on11, en i

= I

we obtain

i

and the resid.ue

of

d_

67.

The right-har:d and

0,

sicle of

this

inequality

s = i2, 3, or

when

4

is

equal

to

l_,

s = 0

when

oi..

so that

u ( * 2 , 1 + d 2 )1 , g t r l + u ( d l + t ) Suppose

d,

-s

ancr 0 (

mod li

u ( * r , i + d " )) 0 ( h )

)

0(h) + 0(dt+t) + []i(si-5)l

v(x2,i;ar) >0(k)

s = 0r

Lhen if

or

1

we have

+ [ r r ( 5 i - s - 2 ) ]) O f r . l + I + [ t ( s i - s - + ) ]

s = 2, 3, or 4

and if

s (4

,

we have

+[L(5i-s-2)] )Otrl

+ 0(dr+r)+[l(si-6)]

Hence

(4.2.r7)

, ( * r , i + a _ ) ) 0 t t ) + 0 ( d 1 + t ) + m a x i o , I r r ( s i - o )] ] -r-,*l

If

we now define

A^

as follows

Ar=0(k)

andfor

f I

(4 .2.I8)

A

-

0,+I

2x2I,

Ao*0(d0+l)

,

ct

even,

I

then we haver jr,rst veri-fied. the beIc,w.

od.d,

:= {

i t no * 0(d0+k+l) ,

stated

0

Br:fore

first

two cases of

we can prove

this

a )

satisfies

for

Lernma(4.2.2o),

generar

o

rve need oni

more Lemma.

Iemmil4.2.19) .

llor

-d + 1

sd o

t,

do

the following

( 2 0(d^+1) - 4, l c [ I ;>l u I

cx

odd.

ct

even

inequa1ities

I

\ 2 0 ( d , + k + 1 )+ k fr

where

0

is

d e f : L n e , Cb y

(4.2.13)

and the

4,

Uo

are dcfinecl by

(4.2.3) .

r,

68.

Proof. .Suppose Ct from Lemma (4 .2 .ll")

isoddand

d^=5r*s

2 0 ( c 1 o + 1= )

l

l

from Leruna (4.2.1I)

=

where

the

Proof .

q

i

f

,

a re

=

O

c

' L

where

=

o

?

' t

r

?

J '

l

n v , :

0(sI

,

da f i nort hru (4 .2 .LB) .

From (4..2.16) we have that

We now proceed

s

i f

'e (xo,i+a 2 oo * *.*{ o ) , t2l;91} -A-cx -

2|(5r+s+r)

5 [ t 1 * " I, - (\ r5! r + s ) - 2 e ( 5 r + s + l )

>/ -4

( 4 -2 -2 r)

then

we have

0,

L e m m a( 4 . 2 . 2 1 . O ) . p o r

-

,

d,r*k=5r+s

5 d ^ , , ,- ( d - * k ) - 2 0 ( C ^-,- +- k' + l ) cx+t_ 0,

2

[ - ' -

I

isevenand

-(5r+s) -

si+]

_

then

O(s(+

rqe have

tuo*r-ds-

Nowsuppose o

where

by :lnduction

on

o .

(4.2.21)

is

Suppose

r. ) ) - A II m a x t u ' Cr " l ( x' l i , r l

true o

is

for

0 = L

od.d, and

' 5i-6" ' , t t

We have

xcr+l,j'|dcr+L=

*c,i ,1,

ai,j+do+r =

x G , , i + d oa i + d o , j * d o * r rl,

,

so from Len$.ia (4.2.I9)

it

fol_lows tJ:at

u[*o*r,:i+qi> *r) { " I t , r * % ) + v ( u i + djo* ,q * J ] Ir'i tlll

i o o* * ' * { o , l \ 9 }

= ,l,rl ifo * *.* {o,t ff I'I

+ - f . z ( s r l + a o (*ir+l -q l - r l)}

} + t ,.(si-i+s O*r-6-r)J}

:: ,5 + [ \(51 + s%+r - do - 2)] > > ; r o+ [ L ( s j + z o t % n r l - e l ]

= : ; t r + 0 ( d o + t-)t l J f

=: rb*:_ +

tI# t .

Further,

v [*o*t,t*do,*t) ] min {ao * v (ur+do,l }, { + du,+r) *i,r{ao* t }f]

+ [ ! ( a _ i + s a o * r _ a o]l ]1 or

(4'2'22) u["o*r,r*ao*r) )min{{ ao + u(*u,uo*r-), d o * d o **,2 ) } , T i : i r ^ o * i - s - : - q ]

By Lemma (4.2.1I) 6(d^+1)

,

+. 4

[---u---]

5d^ + d- + 10

=

J--e-----!L-l

d

=do*tfl

so from (4.2.L5) ii: f,cllor.vsthat the first than or equal to

term in

+2=

dq,*d**t*2,

(4.2.22) is greater

a o + 0 ( 6 ( c i o + 1 )) = o o + 0 ( d o + 1 ) .

The minimum of the lettter L e m m a( 4 . 2 . 1 9 ) w e h a v e

term is attained

when

i = 2

so frorn

70.

u ( * o * r , r * d o * t ) ) m i n { A s* 0 ( c 1 o + t ) o , o * z + [\(z * sao*l - do)] ] ) : m i n { a o+ O ( d d + t ) , A 0 * 0 ( a o + f + I } ) .= .Ao + o(do+l)

= A.,+L

Hence, -6

" o o * r -+ * t * {ro ' 't5l f- -i 9 1 1

,v, [f * . .o * l , j * . i o * ' J

" Now suppose

0

j-s r:ven,

and

u ( * o , i * a o= J oo*

*.*{0, ITt

i

We have

xo,+t,j+dcx+r =

*o,,i bi,j+do+r ,lr_

so from Lernma (4.2.19)

u [*o*r, j*do*r) 't

it

= rl,

follows

x c x , i + d ob i * d o , j + d * + l ,

that

]']i

{u (*o, i*ao) * v [bi+ds, j*uo*r) ]

Tll

{ o o* * u * { 0 , f T l

lll

{oo* *.*{o,tryl } * t r(sj - i - 5do*r_-ao-r2

To do this

divicles; It

m

,

the

first

we wil-tnon-zero

can be shown that

the

: ,A

i,Ij;a] f

The following

LenrmeL is; an improvement

L e m m a( 4 . 3 . 2 ) .

(4.2.I5)

on

rf

j = t+l 5

v [' * * , l >

O(i)

if

0 ( i)

otherwise .

1rl"

.'] =

v [*.

L t

.

rhen i = 1]

or

1 1 m o d2 5 ,

)

Proof. For

L>I,

define

We wiLl

now :;how that,

(4.3.3)

t4 - 5- n - .4

=

(4'3'4)

t5.,-3

=

(4.3. s)

Ms.,_2 =

(4.3.6)

M5r,_t =

M-

=

'

:,A

i,[ *rt]

l- ftZS,-rn + 50n3 - 305n2 + 17Bn 24 ;

(25n3 - t9n + 6) ,

,

( Z S n 2- 5 n - 2 ) ,

5n - 1

and ( 4. 3 . 7 )

M. = m

I

24),

first

73. From {4.1.8) it

can be easily shcwn that

(4'3'B)

M5'+4

(4'3'9)

M5n-r3 =

250'5.,* 15t5,.,-r *

M5r.,n2 =

1 5 M r r . ,* 5M5n *

(4.3.1-0)

=

(4 .3.11)

Ms'+I

=

(4.3.I2)

M_ _ 5n+5

=

It

is

easily

verifie .

4 L I

A

for A

I

A \

\.i.z.Io/

surficiently

with

and

0

,

.

we a1r:eady have mod 24

large

k > 12,

k = 2 4 r + 1 2

= 1 then

for there

CI">L. exists

some

84. l: = w5

where that

+ v

0 some integers

for

?lt

*

satisfy

we have already assume

2 0 ,

o

g* )

* 0, , wr V

1 ( w ( a ,

shorv' the 1.

0 ( v . i 5 0

statement

suppose

is

0,*

is

true

=Q

for

sowemay

odd.

F r o m ( 4 . 4 . 2 . t w e have * --

,t

''

k/q*

tI

l

't\

- r / / -" ;- t t -

-

l

L

*

d a'l ( r + t ' )( s * ' ' \

_

.

l

\

r'r - 1

L

J

'

r^v * f,

*+]

r-_ n f, 5 r +l (

=

L_

r

- ' l

)

*

l

* - - L ) / 2- r - I +(S*

-'l

A

-

\ / )

av -l

L Z . J

n:k f,

since

,o-+1-1= (24+r)s0 *

or

*

r

2

+

)

^ , * - 1. , . f l _ 1 - t ) -= L l ?z . ) t r u n --t-

-l

ar

(2-w) 50

I

*

+1 , -L JR c

* \

t

mod 5

L

r

{2 - w) , l '

=

(50 -l)/2

-0* .v > :_1.

=

d

* c l *+

l

then

-

I

, _

moo5

*

-5c ( (so -r)/2 - v - 1 1

for

CX

Hence the

st,atement

is

true

for

o.)-d.* + 2

o*

odd.

then

m o aS .

-^,

.

86.

Now suppose

0,

Frorn ( 4 . 4 . 2 ) w e h a *

-

d_x,, ft

-

-.

l

(t L') " - 1 )/ 2 4 - r k ( s * ----;=;--,

F I

II

_t

I

t-

f

*

f , - c L + 2l(5 I)/2 5r + i : i '+ :t I n ,

a

-

5

-

-r

-t - l

*

J

iI

r Y A Ir t z . s * + ( s * - r t / 2 - r - 1 -rl

|l

.

*. / ^ + l

5

mod5

I

1 rJ [ p - r l s 0 + ( s 0 - r ) / 2 - v - 1 J Z L J

Now,

if

- 1{ -

E

w = 2

v ) -

and

t -

2

- ,on+l 1(z

or

w=3,4

then

* w 1 5 0 *+ ( 5 0 - L ) / 2 - v - 1 (

so that fo,*+l +1

=

do**l *1

z

moo5

and

> I For the other values of

w

for

and

u2a*

+2

we consider

d * av

From (4.4.4) we have

do*+k

t

k(so - L)/24---F-.-J

r

I I

+K

* * 5a Qqr + 12) + (r + L) (5s -

L

1 ) - r

f,

* l'r2 c* 25r + l _ L

+

t

L)/2 - r - 1

l

f,

m o d 5 .

l

o

87.

i

+ (5

2.54

*

r

,^ \z -

I

t

-0

wi.3

+

mocl 5

L tuo- r)/2 - v - 1 I ---

(2 - w) +

As before

we find

that

mod 5

I

(

maA

)_d

we have

* rr

( ' ' I d O * + k + l - =- Jl ' Ll '

f- 0 *

w

-

l.

and

t(

( s o-1

n t

* I

!

W

_

I

for

w=2

for

i f

w = 2

Thj-s completes the proof

(so -t) /2 - I,

and

,, (

(s0 _ 1 ) / 2 _ i

This

24m = k

best

Now So from

possible

mod 50

unless

log_(k +_24) l.og 5 (4 .4 .3)

* 1,

or

a n d ' : , 4. 4 . 5 t

some exact

w=

for

13

for

we have

4 a K . L

' . A

|

|

-

t

a

l a

|

|

odd,

,

even.

( 4. 6 .r ) -'1 t K - l . 1 I I

l

-

z

l n r

+

|

l

|

L ,

t )

of Lemma (4.5.g) .

We now determine

If

*

olcr

* v((s0-t)/?-r

and

-

q^ l-l tu

Hence from (4.5.7) we have ) t

f

*

I [ 1,

A

\ / )u

t,U.

In

fact

from (4.5.3)

Hence for

a>

2

we observe that

\$.f,.

/J

(4.6.1)

holds

a> 2.

for

l-s

0 ( k ) + 0 ( e . , )+ I 0t.ol , * P 2 tffl so tl:at

cr

iI

Itr From

i )- Z

f}{f

a

true

' for

0, = L

,

A2Z,

90. We have

:1[lE]1 = r*t1!91f1 Now suppose

o )

3,

u(*o,r*u )

g

is

,

ror i)z

odd and

:= t#r

,

u (*o,r*u ) We have

.

so

xo+l-,j+do+l

xo'+1 j ,

=

i ,1, "o,

oi,

j

= ,lr.

xct, ai+l, i+r i

u[*o*r,j*do*rJ * . ' ( " r * r , r )] " T]l {'[*o,r*r)

"T]l irryt

+ * . * { o , t ! # l } + t b ( s j- ( i + 1-) 1 ) Jl

"tt?'t.tTt

= r*t#t

.L5i!.1

- 3J = [ 3 -(cx+11 + m a*{ 0, t3:qr ; Now

uz,r= *12,3 =

104 x 5

.u,ff*-o * r , t * u o * a\ ) = -v.[r* a

,z

since

o

is odd.

so

uz,y) = u(*o,r)

* ,(ur,r)

= 1*1-19::l = 111&$__:1 si-nce

u ( * o , i +a1i + r , r ,- ") t # f

+*.*{o,tfJ } -f?

)> 3+[Sf This

completes

Similarly

the p::oof

k = lLl

for

of

= 2+[]..1*3ll

Lemma (4.6.4)

we can obtain

ror

.

for

cL )_ L ,

= z t$r v(*o,rnuo) and

u ( * o , r * uJ r z t $ l 1

a 1!i:s1

"r-'*c!

ror

i 2 z ,

so that if

24m =: 1l rnod 50

and this

For

is

k = 17

if

and this

24m i

is

best

n_rr(m)

= o

mod 52[@+tl /z)

possible.

we easily 17 mod 50 best

then

poss"ible.

find

then

that

tr>_rr(m) = O mod 52

i2z

93.

INTRODUCl']O}I. Winquist

[17J

el-ementary proofs

(s.1.1) In

this

Our proof

Swinnerton-Dyer's triple give the

product

paper,

case

||q

14, 26. Let

which

using

p = 1I

and

r

I

Suppose that

?\

- t

p

non-negative

integer.

eguation obtain function

of the

(5.1.1)

section e'l eventh

gil_ven

1 can be exten,Ced to in

Atkin

no more than

from Atkin

ano ilacobirs

and Hussain

Theorem,

due to

t s I

we

Newman Lr3l,

for

S 10.

r

is one of the numbers; such that pr(o)

t 1p+1) = 0

a s zero if

0,

A

zr

A

1t

o,

6,

(mod 24) . is

not

a

Then ='

(5.5)

easily

we give

order

due to identity,

p (11n+6) qn .

I

,-rr(r/2)

follows

R.rmanujan-type )l n)0

Chapter

using

forlowing

and defin"

Pt(nP + h) We show that

the

ideas

be a prime > 3

A = r (p2-t) /zs

of

we prove

of

Let

rn

methods

proof

L .

\ J .

nave

mod 11.

depends on an identity

iderrtity.

an erementary

Thanrom

=0

we show how the

(5.1.f).

L .* I

and Swinnerton-Dyer

of

f(l l n +6 ) chapter

derive

and Atkin

nr{n7P1

from Theorem (5.1.2).

an elementary rine

[

stated

8 ],

deriva.tion

of

the modular

from which we are

be1ow,

for

the

able

g€,nerating

to

J_Ltr

>4.

rl'horrrom

Iq \

J

.

I !

.

?l

J

'

.

qdr]l [ t ( 1 1 n + 6 ) q n = 1 1 q 4 ( r r 3+ 8 B A *+ r r * 2* 1 1 s * )

(s.11..4)

n)O

* * A , B

whet.e d A t l

h a d

t

n, -rI2

are power series

q

in

with

integer

coefficients

h l t

( ] O A *+ r 1 2 ) r ( q l l ) s =

I n.trln + zs)qn

rt)-2' and

(14B +It2A

We note here that

5.2..the

f n this

residue

( L . 2 1. 2 ) .

of

*

3

(5 ..I.4) is

sectirrn the

l'l

7

^

+11")E(q";'=

n

I p"(t1n+35)q n)-3

( 3,25) in Fine' s paper.

we split

Euler,s

ex.ponent (mod 11) and obtain

These are

r:ontained

in

the

n (cl)

function

following

relations

;rccord.ing to analoqous

Lemma.

Lemma (5.2.1)_.

p ( e ) = u ( q r 2 l )- e o v;here

Qn '

=

Ql=

t j _L2In-44,,, ,., \r-9--^L2In-77, i Tl- il-ST-,,'.t (t-qt21n-22r{t-nl21n-oo,

TT

n>1

"'-'t

Tf "=t .ffi

Y1

1 ) 1 n - ) -) - )

(r-q-l-"

1 ? 1 n - Q- Q ")

(1-q--'"

il n2l"

r

(t-n12rn-ta){r_ntr1n-110, (t-o12tn-tu) {t-ot'ln-66,

Q 2 =

Qn =

q e r - n ' e , - n t u n n* q 5 * J e ;

(t-n121n-") (r.nt'1n-88, (t - n l 2 t n - t t ) t r - n t ' t n * 1 1 0 , (t-qI2ln-tt)

{r-ot'1n*65,

(t-ql'1t-t')

{r-nt'1n-88,

'

f

,

,

Eo

qrti

and these

qfv

n n 2

* zn22erene, ,nttn, e7 = o a|a,,* a33af;

-

voYl

n"t,rn| * ,3 * ,nlleoeren ,r"ene, = o

- -. L, .- 2' 2 2 Q o Q z*

( s. 2 . 2 )

q

z

z

^ 1 1 zq Q L Q 2 Q 4 - 2 Q o Q z =

,"t-n 1 . a:, 2

QoQa - q

)

IT2 11()2 + 2QrQ, = o Q r Q n + t ' - 7 - 2QoQrQT

ttt'orof,* ol

aiaz (s.2.3)

- 2QoSzQt-

Here we assume Jacobirs

can

r-t

-

vn

YnYl

product

triple

'

identity

(r + tq2n-t) (t * a-ln2n-t) {:- - o"1

t t n)L from which

'l 1 2n'-^

I

Qo Qr Q2 Q4 Q7

TT

o

=

c0 I.)

n

n

deduced

L^

c

e(q)

o

)

n L ( 3 'n - - n ) ) (-r) q"

r

E( e )

(Eul-er) ,

2*rr)

3

::

I

L(n t - l ) n ( z n+ r ) q

(Jacobi)

,

n)0 and lriatson' s quintuple:

product

identity

(r+rq n ),(r + t- 1qnt) ( r - .' n"- t)

TI

(l-t

-)

- q *) n"- 1 ^ )

?r

(1-q"1

n21

= T f tr-t3 n 3 n -1 ,(1 -r'-3q3t- ' ){ r - n") n):1

(For

a n elementary

Write

E (q)

=

proof

* t- l ff

tr - t3n3n- 2,( r - t- 3q3t - t) tr - - ntty

n)l

of t h e q u i n t u p l e p r o d u c t i d e n t i t y ,

see Bailey' t0 J.)

10

I

I

where congruent

contalns

E. a

i

to

Since

I (:r

and

E( s )

mod

2 * - r + -

L t

f

thosb terms

af

!F

f\ n \ Y'

i n

u'hi nh

fha

nnwor

nf

II. 3 , 6 , B , 9 , 10 mod 11,

E o * E I + 8 2 + E 4 * U 5 * E 7

E 3 = E 6 = E B = =E , = E r O = 0

I:'

uo(q)

-r) ( - 1 )-r q\-( '3-r-2 ^ ' +

r

1 +

I r) 1

r'

F

)

f\ - lL )' -

L

r

tf

^Y " " '

l-2r*'l

( - 1 .l

1

b(:r

+r'l

L

121 v ? O, 4 nod 1l6

b ( 363n2+tIn)

In

q

( ( - . 1 )r n t 3 r 2 - r )

T

+

r>l :r=U, I mod 11

F

-r

r)l L ( 3 1 2 + r )= o nod 11

L ( 312-r)=O mod 11

1 +

/\ -- -1L \l

\ L

r,

\ ( ' l 6 z n 2+ 2 5 3 n + 4 4 )

, -f, , l l n * 4

Lf

t

,

Y

-@

'lr

( T I

-;t .: - t z '' ll

n

^--

JOJ / t - . t t a t z

n

\

o

t l

J. - . d 1

I

Y,

l

zz )

J

l

n l

r

^

t

Il a 2 - Il

qz

-

n

z)5

)l

Jb.J)

\'1

)

-O

- T - T( 3 6 3 n - 1 8 7r 363n-176r363nr l l [ r - q J l r - s J [ r - s ) n)1 + q

22 -rr I l

(.l63nr 363n-55r 3 6 3 n - 3 0 8r l - r - q J t r - q J [ r - s )

n):I

T-r il

(. . L2Ln-22t rtl-+q

/

l l t + q\

121n-99r r. r

l l l\- - q

/

242n-I65t rr

/

\

t l J -3 - q

2 4 ' . 2 n - 7 7rt/

\

t lf l - o

12lnr

n>1

=

Tf " t n t " l n):f

= n ( q 1 2 1T1f

7, (t-q242n-nn ) ( r - o t n 2 n - 1 9 8 ,( r - a 2 4 2 - r u u ){ r _ - o ' n 2 n - 7 ( t-nl2rn-t' , r _yo l 2 l n \r

n):1 ( t - q l

Similarly

n, (u)

21t-

) { r-r"ln-9e,

7, nn ( )/ \ rI _ \o1 I r L n - 7 " ) {r-nt'ln-99,

we obtain:

- en(q12:r) TT t l

r, , r_o121n- ) e_ otrln-99, \ +

,

a

\ 4

a

rI>1 ( r - q 1 2 1 t - t t ) t t - n

"z

(n)

ua(q)

= - q2'(nu'r)

Tf n)'l

- - q 1 5- _u .( 91 2 I . )

.ff

il

n)1

a- (q)

q 7 E (s1 2 1) Ti n21

,r-o121n-tt) (t-o (t-nl2tn-")

{t-n

121n-110, 12In-66, I2In-88,

(t-ql21t-tt) (t-o 12ln-110, {t-nl2tn-tu) tt-n (r-ql21t-")

{ t-n1ztn-

tt-n

12ln-66, I2ln-88,

77', nn ) tr-rt'rn-

I

97. Now,

E- (q) 5

I

-

\(Zr, -r) =5 mod Il

t-r)

T r ..f t-rl I t=2 mod 11

t'

q%(3r2-r)

L(3r2-r) q:

@

r l

(, - ,r .l I I n + 2 q h- (' : O : n 2 + t 2 1 n + I 0 )

i I

(, - r, ,. D L1q- I 2 \)L ( 3 n 2 + n )

_ r I2Ir E[q

Combining

these

results

)

we obtain

E ( q ) = p ( q 1 2 1 )[ a o - q Q r - , ' e , - u l s e , ** where

the

crearlv

Qi

are

defined

as in

the

'd 1 r

statement

^' Yr Y \ j )l

of

the

?F

E.-\

,

Lemma.

Qo gr ez ea e7 = r

Now,

n(e)3

- ( u o - r -E f + E 2 * u 4 + E 5 + E 7 )3 = t e j + 6 u 2 u 4 u s* + 6 E o E 4 r i 7+ .zrlu, + ( E ; - F 6 E 1 E 4 E 7* : " f u ,

+ 6 E o E s r T+

?r. Tr-\ z a

+ (:norii * zuf,u,* zf,u, * 6u2u.1u7 r I

+ tnl -F6EoEtE2* :uoul * 6"zu:;u7 I ,

+ (snoul * nlu.,* * =ulu,* rufr"n

ar r v D l u _ u _ t

a r h - \ J u ^ ' _ l

+ f

'. -oL. " 1. 2

+ (errnl* ruj", * .,rrl * 6uou:L"a ' I

+ G] -F 6E.E2E4 * zu2run* 6uou.Lu5

\

I f , I

af, vD

*

rllt ":i'

r r aDFu-t

4

'

. h - -

I

J D r D a t

)

\

I

\

+ (3Eirrs* =uf,u-* runul* 6Elu:zu4 '""o"2"5' r A E . E . F \

+ :lr.2ir4 * .uou! * zurtl * 6u1",2"s, |

/f

+ (:nrnl * tu'ru, + nlx, * 6EoE,rEs '

r + tul + 6E1E4E' * t rrf, * 6uL"2"7 '

r

f

""o"I"7, ! ! ^ ! ^ ! _ /

u z t a

?r: l'- \

J D ^ L r t

U

f

\

oo

t - u n (2n+I)

I

2 +n)

1 rl n

q'"'

n)tJ

Since

l(n

rI

n\

a

t L t

a

L t

t

2o'

a"- 12

(5.2.4)

""0"1

ao. '"o"

a rr"r2 ,2 '"

^^2-

.r 2r,

'a".2.," "o"7

" l=

-- --.-1r 5

obtain

-L

?F

2t 'a'. "L"7

t

)

+

t

6EoEzts +

t

r' F 4 t ) I

""1"r"

=

n

aI

+

? n-r JD^ b-

z

)

6E}E2E.7 +

and

5oc,uau5

E2 = -q2ezE(,r12t)

E7 = q7 e|u (e12'1)

we

(5.2.2\ .

The proof

of

ttre foJ-lowing

Lemma is

cornpletely

analogous

L e m m a( 1 . 2 . 5 ) . 'l 't

Lemma(5.2.5)..

10

ll

i=0

If

'I

n (to--q)

u

+ 6"o"tuz + 6utuz"s

Eu = q5E(q121)

B

AI.

f

a

+

o ' t ^ 1 2 1 r) , Y,r-\Y

r

6E^E.E , u r 4

T;!-

4

l

EO = e ( q 1 2 1 ) Q o , E f = - q Q l E ( q 1 ' 1 ) ,

Substituting n

'a" o2 o" 7. 2

z

o 1 +

ao, ',2 L

*'

t

ao.2"' z . +

we have

) 3 E. ;t E _ f ,

+

2

U 5

t 5

11

v t

--

'

l

r . L t

r

I r''.I12 n tq-^| , . ' l? 1 . rr (^--* t

w

J

' f

l

L ,

then

to

that

of

oo

5.3.

rL appears tl:at

sufficient

+ O; = 0

l(lln

this

mod 11

srectir:n we will

Qua ' Q 1 , _Q r , z Q n4 , Ql . , J will

(5.2.2)

in

and (5.?..3) are not

to prove

(5.3.I)

In

the identities

derive

due to Atkin

further

iclentities

involving

and Swinnerton-Dyer

ena]:rle uri tcr nrcr\,o (5.3.1)

later

n > 0

for

.

.4 ),

[

FolJ-ovring Atkin

vrhich

and Swinnerton-

D1'er we define

p(a,x)=

(5.3.2)

for

Tf tr-.*t-1)(r-a-t*t)

alo, l"l.r.

n>l First

we need some l?roperties

p.

of

L e m m a( 5 . 3 . 3 ) . (5.3.4)

p(a-I

(5.3.5)

P(ax, x,r

(5.3.6)

P(a-1,:r)

(5.3.7)

p(a, x2) n(x2) =

x,x)

p(a, x)

=

=

|

-

a-l .p(a,x)

|

-

a-l

p(a,x)

,

(-r)n at xn(t-"l)

i -@

Proof.

(5.3.4),

Jacobi's

triple

(11.3.5)and (5.3.6)

product

-l

TT r

(1 - ax(x-)" *)(1 -

?n 2 n - 1* ) ( 1 - x - ' ^ ) x

=

n21

ll

from (5.3.2).

easily

is

identity

. 2n.-l ( I - a x - " - ) ( l - a

follow

(ax)

-

x-..)(1-x-")

c

o

r I ( - 1 )

n

=

I

s

n21

) P(ax, :I

and

Now suppose If

d > 2.

b>l

e>l/

and

e>I,

c)_ 1

ancl d,zl_,

respectively,

b > I

d > I.

and

c = €: = 0 .

We may assume

we haver

I o tB t'6d = I ou6oy. = P( Y,z) , for

p .

some polynomial

From the

third

equation

so that

in

=: -

d.g2 i-f

b > 3

I ot8bo

=

So we may assume (5.4.3)

d = 1.

we have

o26 - 2y6e - y2 - zg.

,

we have

), too2l oa-l ub-2 u

= - i , o " * 1 8 b - 2 d 2- 2 l

- | aa-lgb-z.rzu oa-lUb-2rd2e

- 2 | oa- 1ub- 1u.

= - tr o2ra+1ub-'* - I o"-tBo-'"r'6-rl o"-lgn-ru. rl oa-2ub-3u =

P(YrZ),

W e m a y a s s u m eb = 1 , fron

( 5.4.I2)

for

some polynomial

s i n c e i f

p.

b = 2 r d = l r c = e = O

t h e n

we have

6 a + 1 4 + B

=

0

m o d l l

a

=

0

mod 11, which is impossible.

Il3. For

b=d=11c=e=0

a = 3,

'the statenrent is

Now suppose suppose c22

c > I

e22

So we may assume is

We have the

I o"yudt

=

p (y ,z) ,

=

I o"Bt,j"

=

p(ytz) ,

c = e = I

.

In

Each i:erm has only'two following

this

case we find

d > l. b = d = o.

that

d = 9,

factors.

table:

Terms with z

6

C

7

0

L

^.t O

5 ^ 2 p

6 " c L o

IO

we may assume

=

Degree

9

.

ancl

impossible.

CASE (II):

8

e 2 I

b > I

Hence

p

some polynomial

which

for

contradiction.

weharte

I ou.f"r" for

and

true

rvhicfuimplies

p

some polynomial

If

modII

wehave

I ou.rtr" for

a=3

a + b * c * d + e = 5 = n + 1,

but

we have shown that

If

wefind

two

^ 2' 4

Y

0

^ 5 2 F Y

Y

5 " 2 0

^6

6 y 0 .

K

K re y

^7

r t K

F Y

z . *

"7 Y 0

a 0 l J

^8 F Y

3 ^ 6 cr, o

K t r

0 , b

^ 4 6 F Y

Y

^ 7 .f 3

7 3 Y 0 ,

7 "3

^3

K P

v

6

" ( 0 3 6 Y o ( .l

"o o

factors

2

4

N

" 2 ^K 4

e

" 5 2

s

" 6 ^

t r ^ v

Y

5a

2

t

6

r ^ 7 R c

^6 o" 3 F ^zt

o

"7 ^3

e

d

3 4

Y

t r d

e

t

5 6

5

Y

114. We will

now show thtrt

each of

the

expressions,

I o ' o n , I o t g ' , ,I o u o , I o g t , I o r B , I o 3 6 6 , ' I o n B u ,I o ' 6 t , can be written

p(y,z)

as

for

,

p

some polynomial

We have

Io'on =

[ i o o 2 ) ' - , I o 6 6 2 e -2z l o 8 2 6 3

= since

P(Y,Z)

for

,

from Lemma (s.4.g),

expressionsi

are

of

( I

p ,

some polynomial =

oO')'

d.egree 6 and. they

16 ,

have

and the

terms

with

at

latter

least

rwo

ttrree

factors. T ^.6o

Multiplying

6

r

=

100

=

loy-

the third

l,

bydefinition.

(5.4.3) by

equation in

o4

o4y2 * zct4ge* o6d * o5g2 + zcr,4y6e =

we obtain

o ,

so that

-

IotB'=-[o66 = -

since are

the

of

z + P(Y,Z),

last

degree

Similarly

the

three 7 with desired

_ \ 2

which

6

.

terms

)

equarion

e.

the

firsL c.

I

q

is

Multiplying

are either

for

o,'9'6 + in

equation R )

).

2].oa8e p,

somepolynomial

consisting

result

+

third

for

expressions

B"\'+z!"e the

Lony'-

zlo;yo,

of

degree

least

less

three

follows

than

factors.

from

q ,

?

(5.4.3)

z$'y6e = multiplied

(5.4.3) 6.

at

oB7

aB' +

in

of

)

o " B + 2 o " B y e - r c t " 8 ' 6 '* o o B 6 e ,

by

+

o, by

oUB .t^2

85 we obtain

2a'b,Y =

e,

7 or

115. so

that

.

r

(5.4.17)

we wil-r

tI

,

-

each expression

c,an be writLen type.

terms with

it

at

o63ye =

I

has the least

on the

as a polynomial

Now,,

since

A

6^2^2 | = =- 2 [ o o B v r- l o b o - ;

now show th;rt

(5.4.17) desired

B.

).o"9

three

as

6_- ? r 7 o o B 6 r ' t- 2 , I. o r g r ,

right-hand. y

in

pl (y,z)

same degree

"

for I

and

sj.de of Z ,

of

the

some polyrromial

oBB

and it

consists

of

factors.

We have

(s.4.r8)

( [ o 3 s o J 2= [ o 6 B 2 o-' , 2 L o ' , B ' * z I o 4 B v 3 e ; 2

From ( 5.4 . 13)

it

fol-lows

that

q _4^ :3.2 r 2^ 2"2 r 2 3"3 l o . F Y $ = - 1 0 5 Y 0 - I o y o =

P

2(Y,z)

for some polynomial.

,

since each expression has degree less

n

than 9, the degree of

Z , I o8B

N o w , f r o m L e m m a( 5 . 4 . 8 )

( [ c , 3 e o2 J -

I o68262 = p 3ft ,z) ,

and

Now, from (5.4.I3)

1--2y+y2

[I"yr')'=

for somepolynomia]- n3

vre have

b^" !l r l c I F o e

2^6 r L 0 F Y e

=

= - I eF^ e6 =

'

-

P 4(Y ,z)

r ^ 5

I b y o e for some PolYnomi;rl

,

Similarly,

- ^ a 'B-y I

=

Ps(y,z)

,

for some polynomial

Hence,

r )

=

p (Y ,Z)

,

for

B cr-B

some polynomial

nq n5 P

116. l,lultiplying

the third

(5.4.3) by

equation in

oy2d5 + z*B65e * ot66 * o'B'6t

0,65

we obtain

+ zoy66e =

0,

so that

-, I o'ou = - I o2B'05 =

since than

P (Y ,Z)

each expression 9 or

is

of

This

on the

to

F 73 )o 'y "

completes

the proof

As an immediate

u

Iror each

Y, Z

'

is

either

consisting

of

desired

result

at

of

leas.L three

factcrs.

f,-rr

Lemma (5.4.fI).

j_,

I

the

fo]lowinq

< i

< lO,

Lemma.

there

exis;t

integers

is

=

c x + B + Y + 6 + e+ 1

q-E (q"') o,

l-ess

,Y2 + sY + tz * u ,

E =-9!4"' where

deqree

are clefined by (5.4.7) .

(5.4.2)

Proof.

side

such tliat

H(Ei) where

the

consreguence we have

Lemna (5.4.19) . tr

terms

obtain

of

p ,

some polynomial

right-hand

deg:lee 9 with

a n d

f r s,

for

,

We can arg'ue simila::J-y 4 ^6 r 1 0 5

- I o y ' o u- z \ o 8 6 5 r : I oyo6e

3, y,

(5.4.1).

6,

e

-

^ -i (c + g + y + tj + e + 1)

are defined by

,

Now, -i t,

is

a sum of

of the form 1 *^ a o b . , c ^vd - e c: t, I L

- c ^ ,ra- '.8o"by^".6c *r e d ^- e oa-.b"c^d-e = + B *y"6"e*cr-

+ 0" aeb oc ^Fd Ye w h e r e a + b + c + d + e < 1 0

a"b

^

I

+ y- * 6 " e ' o " B =

a b ^ c d " e + e o F Y c

f

^

expressions

1.17. From the

remarks

at

the

beginning

the proof

of

Lemma (5.4.11)

of

we ha,.ze

that

H(E-) 1 ^aob^,c^d,-e lupYou

is; a sum of where

,

d,b,

a + b + c + d + e < 10 .

5.5

In this

r,

of

c,d,e

the

form

s a t i s f i e s ( 5 . 4 . 1 2 )a n < l

Hence from Lemma (5.4.fI)

H(Ei) = for some integers

expressions

we have

r v 2+ s Y + t z + u ,

s, t,

u .

secti.on we prove a special case of fheorem (5.1.2),

fron which we are ab,Ie to show that n(lln We also

+ 61 = 0

der:ive

the

mod 11,

rnodular

n > 0

for

equation

of

eleventh

order

due to

nirre

I g ]

We define ( 5 . 5 . 1 )

A

where

are defined

Y, Z

=

- 1 6 - y ,

we now introduce

the

series

of

of

powers

the power of

q

=

2 - 2 ,

(5.4.7) .

operators q

is

by

B

and simply

congruent

0 < i

Hi,

to

pick i

< 10 ,

out

these

modul_o 11,

which terms so that

act

7 ,

E(S)i

Now, il (q) =

-

4

C

q

e(9

1 ^ i LZL.

)

QA

' - u

n :1

'7

( s .s . 3 ) Ht : 6 ' )

/ F / ^ \ ' \ rr.r\LrY/

u (ql21) -40, - -H7

q

(s.s.4) H(qB)

-

=I4x-

7

q

( 5 . s . 6 ) u ( 6 1 0 )= -

of

^ - 0 x - + 0 x - + 0x - - 1I'+

1r1.8

= -9x

( q - - - )g

)

121.10 s(q ) first

powers of

-4

-4

135x

-

-+

-

v

*

= -x

P

-2

- ^z x- 1

x

-,f

x - * 4 x + y4x

From Lemma ( 5 .4 .19) '

7

H ( g ' )=

)

-

l

2

vL

t

?

4

-

/U

_

-

e

/

e

,

1+

.

' t l -

5

4

-3

A

5 ^ z _ 4 - 5 x - 8 x x -'7x +l-1x

+

I TJX

x

^-2 - x -1 + u x + 1 4 - 2 x -

16x' - 18x- + 46x a

,

-t

L56€:x-2+ 3015x

{

el_even cerms o f

r 6 I rlY

Y

O + ...

= 0 x - * O x - + 0 x - + 0 x t +

- lox6 + 12x

(s.s.e)

l

* = qll

L

(s.s.8)

I

)

1 ) 1

the

(s.s.7)

lgl+595x+....

-R)

(E (q)

-qn rn " " r1t1 6 1 r ./ ^ \ ' " r a .1 \! \Y/

series

-

+ zJSx

q - 4 'q* u r ( E ( q ) -)

r

We have calculated

-'l

+IL2x'

-

-r , ( q

( s . s . s ) Ht E e )

I

J

+

3l-x7 +

/

zx

-

,

z + I 2 x - + 2 6 x ' + 5 2 + 14x -t- .107x

+ lo6x4 + r4x5 -

\f,'e have 2

rlr-+sY+tZ+u

6

a

J-bx

i

.

I

'l

for some integers r = 0 ,

t = - 1 , 17

u = -

and

t€t s, t,

433

.

u,

tn

so that J 4 t- 4 s * u

s = - J . I 2 ,

=

_ 1 U l

lFherpfora-

'7

- 1 1 2 ( -r o - A ) . - t 4 ( 2 - B ) - 4 3 3

H(E') = =

l4B + ll2a + rr3

Similar1y.

.*8 H(E-) = r

=

=

s

t=0

s H(8") = r

- 9 ,

=

and

and

- lI3

=

u

sothat

2 rY'+sy+tz+u, 4r - L

+

52r

z rY-+sy+tZ+u,

=

-

I4t

ds required.

r

sothat

- 135,

12r - s

+ u

= - 2423.

4s

Hencel T

-

-9,

s(89)

=

- 9 ( A + - 1 6' -)

t=

=

1566

99, s=-16j4, rurn(- t6 -A)

u=-10037

and

+ 99(2 -B) + 10037

= - gA2 +- 13864 - 998 + 1t4 -10 2 H ( E * " ) = r Y - * s i Y* t z + u , r

=s=t=0

and

u=114 r

Theorem(5.5.10).

Suppose

and define

as zero if

pr(o) nr(lln

+' 5r)

=

F r o m L e r u n a( 5 . 5 . I )

Proof.

so that dsrequired.

r

is one of the numbers o

is not a non-negative

-r (- 1r) ft/2) nr(n/11) we have

H(Er) =: (- rr) k/2)-r , H(q-5t E (q) c I

.io

t)

='

(- ,r,

.r.rn .__ = p-(lln+Sr) q--"

t

(r/2) -1

(-tI)

t E(q121)

(' r / 2 ) _ I

,

r p (, n ) q I 2 I n ). t

.,!o-

2, 4, 6, B, 10, inteqer,

Then

121.

or

I^ n)0

Simil-arly

=

n r- ( l 1 n + 5 r : ) q n

we also fj-nd that

I n -. , ( l } n + 5 ) q n n)0

n

(-

tt,

(t/2) -t

I n -r ( n , / 1 1 ) q n r ,eO

ds required.

I

=

I p . ,( n / 1 1 ) q n n2O i

,

= I ^ n) = ( 101 n+ "t s ) qnn 2 1 t OI n .' ( n , / l 1 ) q n

Furthrer,

I

n.(lln + ,)nttt*'

=

e 2 5 ( t o o+ t t 2 ) n ( q 1 2 1 ) 5 ,

+ zs)qIlt

=

(toa + tt2) u(q121)5

n ^ ( 1 1 n+ z s ) q n

=

(1OA + tt2) n(q11)5 ,

n>0

"l_rnr(l1n or (s.s.11) \I

n> -2

=

o* (n)

where

o(qllI1)

Similarly (5.5.12)

nr(lln I n>-3 *

Here 'we note if

fourt.h are

7

( 1 - 4 B *+ r 1 2 A + t t 3 ) "(q11)

,

1/'l 1

B (q-'**)

where

Also

+ 35)qn=

that

A*,

B*

are

we compare (5.5.1r), equation

related

of

to

(46)

A*, GZ =

T h e o r , e m( 5 . 5 . 1 3 ) .

in

B*

(5.5.L2) Atkin

For

p(lln+6)

[ 1]

orB

respectively we find

that

by

* A

respectively

* G" =

,

n ) =0

O , modll

B

-3A

*

in to

F.ine,s paper.

the

Atkin,:s

bhird

and

G?, G3

r22. From Theorem (S.s.fO) we have Pt-u . ,^ ( 1 - L n + 6 ;

=

. ^ ( (n - a ) , 2 1 1 ) 114 p - 10

so that 'ln -H ' 6_ '-(

E ( o ) * "t r = O "1',

mod 1I

Now,

I E(q)

I P("1

n)0

q

It

follows

n ( q)r o E(q)

= =( t - S ) - t

1 nd = 0 + ,t l

.3

_3

-L.

+

11 ,

J.

t' 1 0

+

9to

>z

= 1 1 ,

+

t-10

=-Iit2

where

I23 -

n a= g 3 r . E t * * E i o= - r r 2 , n s = q l t . , i . . . . ,* E i o = l l ( l o a+ r r 2 ) , n e : : g 3 + . g. .i . . * e i o = r 1 3 , =: el . el . ... * glo = 11(148 + L r 2 A+ r r 3 ) ,

,t

"s::

q ; 6 i . . . . * q i o= - u 4 ,

ng =: g3

Ei.

"'

* E ? o = r t ( - g o 2+ 1 3 8 6 A - e e B+ . t t 4 )

Plo=, 6*o qlo* . .. . 813 = rr5 From rstandard

tl

formul,ae

it

fol_lows

that

= IE, = 11,

sz =rlr"'t' = ': ,.1*

5x11 '

i;iEjgk = 112 ,

ta =r.rlo.n 6i6j6kg[ = : Lr2 , t5

=

-11(:tt2-zo)

so

=

- 1 1 l l (1 1 - 2 A ) ,

tz

=

tr(rl3

se

=

1 ] 2 ( 5 x 1 1 2 + 3 B A+ 2 8 ) ,

tn

=

I1(rt4 + 72 xltA

sro =

rr2(lr3

'

,

+ 126A+ 28) ,

- o2 * g x 11B),

+ B x 1la + a2 + tte)

I24. From Lemma ( 5 . 2. 5) vre have 'l

1(l c

-

ol1

=-

f

f

-

ll i=0

L, -

-

1

-

=

o r

ll ,i=0

i

6(tlt-q)

= Itll -F+b-=i=o qt rtt

Hence the

qi

are

the

roots

=

n(e12r)

of

the

==u'o11ltl= t,

u 5 5 "{ n r r t ,

equation

x l l - . r t x l 0 +s * u x e ' - r r 2 x 8+ . . . + r - r 2 ( r r 3 + e x l 1 A + A 2 + r - r B ) x ,*ffi- - ) q E(q 6 = 6O

but

5.6.

and the

We are now in. a position

From Lemma(5.5.2)

-H-(.-E- r* .)

Lemma is

and (5.5.15)

proved

to prove Theorem (5.I.3). we have

n r . , r 1 ;1 2

.;"'Y; ; q 5 5 E( q 1 2 t ) 1 2

= H ( q 1 0 )- r r g ( 6 e ) + s s u ( 6 8 )- : - r 2 n ( 6 7 ) - r r 2 H ( 6 6 ) + 1 1 (t t 2 . z a ) i r ( 6 s l - L r 2 ( 1 1 - 2 A H ) ( g 4 ) - l t ( r 1 3 + r 2 6 a+ 2 l ; )u ( E 3 ) + 1 1 2 ( sx t t 2 + 3 8 A+ z e ) H ( E 2 )- 1 1 ( t t 4 + 7 2 x r r A - o 2 * n r " ) H ( q ) + 1 1 2 ( 1 1 3+ B g A+ a 2 + t l s ) =

1 1 4 - r r ( - g e 2 + r - 3 8 6 A -9 9 8 + t t 4 ) - 5 x 1 r 4 - l t 2 ( t a e + 1 r 2 A + r 1 3 )

- 1 1 4 + r 1 ( r 1 2 - 2 A )( 1 o a + t : - 2 ) + r t 3 ( r r

- z a )+ t t 2 ( 1 t 3 + t 2 6 A + 2 8 )

+ t t 2 ( 5 x r l 2 + 3 g A+ 2 l , ) - 1 r ( t t 4 + 7 2 x 1 y s - o ' * n g " ) + r t 2 ( t t 3 + B B A+ a 2 + 1 t e ) . =

1 1 ( r r 3 + B 8 A* o 2 * t t " )

,

= o,

125. or

( 5 . 6 , 1 ) H ( E - l ) : : t r ( r r 3 + 8 8 A+ a 2 + r t e ) nr, ## Now llrom (5.4.2)

we have

= : +(.q ) I r(r,1qn e

=

out

q

n)o

Picking

those

p'wers

= 6-l^= q 5 E( q r 2 1 )

of

. r 1n+6 ) p(rrn + 6)q* n>O

congruent

to

6

mod r-r

we obtain

1

=

qr.l

.\

q s E( q 1 2 r )

= r r q5o( 1r 3+ 88A+ a2+ r r e) p tq131t- i i E (q-*)

ptrtn + o)qllt

I

=

--

ttq44(1r3+ B8A+ e2 + rln) ta4:#

n /^rrr rz

n)O so that

( 5 . 6 . 2 ) I p t t l n + 6 ) q n = 1 r s 4 ( t t 3 + , r o * * A * 2 +1 r - B * , l , n t t ] 1 1 nio n ( q)t2 We note here that Also,

A

*

(s.s.12).

and

(5.6.2) is (3.25) in Fine's paper. * B are uniquely determined by (5.5.11) and

,

126.

RIBLIOGRAPHY

LIJ

A.O.L.

Atkin, "proof of a conjecture of Glasgow Math. J., e 0967) , 14-32.

lzl

A.o.L.

Atl