We shall obtain the exact formulas for the number of representations by primitive binury qvadratic forms with discriminants -128 and -140

1. Teimuraz Vep¬khvadze Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 1) January 2016, pp.81-83
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On The Number of Representations of a Positive Integer by the
Binary Quadratic Forms with Discriminants -128, -140
Teimuraz Vepkhvadze
(Department of Mathematics, Iv. Javakhishvili Tbilisi State University, Georgia)
Abstract
We shall obtain the exact formulas for the number of representations by primitive binury qvadratic forms with
discriminants -128 and -140.
Key words and phrases: binary quadratic form, genera, class of forms.
I. Introduction
Let   22
; cybxyaxyxff  be a
primitive integral positive-definite binary quadratic
form. The positive integer n is said to be represented
by the form f if there exists integers x and y such that
22
cybxyaxn  .
The number of representations of n by the form f
is denoted by  fnr ; . It is well known how to find
the formulas for the number of representations of a
positive integer by the positive-definite quadratic
form which belong to one-class genera. Some papers
are devoted to the case of multy-class genera. Using
the simple theta functions Peterson [1] obtained
formulas for  fnr ; in the case of the binary forms
with discriminant 44. These forms and some other
ones were considered by P.Kaplan and k.S.Williams
[2]. Their proof for odd number n based on Dirichlet
theorem. In the same work in case of forms with
discriminants equal to 80, 128 and 140
application of this theorem did not succeed and
formulas only for even n have been received. In [3]
we considered two binary forms
22
723 yxyx 
and
22
723 yxyx  of discriminant 80 and
two binary forms
22
1123 yxyx  and
22
1123 yxyx  of discriminant 128. Using
Siegel’s theorem [4] we obtained exact formulas for
the number of representations by these forms. But in
case of the other primitive forms with discriminants
128 and 140 we have to use the theory of modular
forms. In this paper by means of the theory of
modular forms the formulas for the number of
representations of a positive integer by the forms
22
1 32 yxf  ,
22
2 944 yxyxf  ,
22
3 35 yxf  ,
22
4 924 yxyxf  ,
22
5 924 yxyxf  ,
22
6 75 yxf  ,
22
7 1223 yxyxf  ,
22
8 1223 yxyxf  are obtained.
II. Basic results
In order to use the theory of modular forms in
case of the binary forms ( 1, 2, 8)kf k   ) it is
necessary to construct the cusp form  X which is
so-called remainder member. For this purpose we use
the modular properties of the generalized theta –
function defined in [5] as follows:
   
 
 
 




Ngx
N
Axxi
N
gxAh
gh expfp
mod
2
2
1,;


Here A is an integral matrix of f,
S
Zx , g and
h are the special vectors with respect to the form
 xpf , is a spherical function of the  -th order
corresponding to f ; N is a step of the form f.
In particular, if f is a binary form, g and h are
zero vectors and   10 xp , then
   ffpgh ;,; 0   ,
 ;r n f is a Fourier coefficient of  ; f  .
We assume, that
   ffp ghgh ;,; 0   , where 10 p .
 fE ; is the Eisenstein series corresponding to
f (see, e.g., [3]).
By means of the theory of modular forms we
prove the following theorems.
Theorem 1.
Let
22
1 32 yxf  ,
22
2 944 yxyxf  , 






0
16
g ,
RESEARCH ARTICLE OPEN ACCESS

2. Teimuraz Vep¬khvadze Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 1) January 2016, pp.81-83
www.ijera.com 82|P a g e







2
0
h ,
22
84 yxf  . Then we have
     ffEf gh ;
2
1
;
2
1
; 11   ,
     ffEf gh ;
2
1
;
2
1
; 12   .
Theorem 2.
Let
22
3 35 yxf  ,
22
4 924 yxyxf 
22
5 924 yxyxf  ,







0
70
g , 






0
70
h .
Then we have
     433 ;
3
2
;
2
1
; ffEf gh  
       4354 ;
3
1
;
2
1
;; ffEff gh  
Theorem 3.
Let
22
6 75 yxf  ,
22
7 1223 yxyxf  ,
22
8 1223 yxyyf  , 






70
0
g , 






0
70
h .
Then we have
     766 ;
3
2
;
2
1
; ffEf gh   ,
       7687 ;
3
1
;
2
1
;; ffEff gh  
Equating the Fourier coefficients in both sides of the
identities from theorems 1-3 we get the following
theorems:
Theorem 4
Let un 
2 ,   12, u ,
22
1 32 yxf 
22
2 944 yxyxf  . Then
    




 

u
kk fnfnr



;
2
; for
 8mod1u ,
 




 

u 
2
2 for 2 ,  8mod1u and for
3 ,  8mod3,1u ,
0 otherwise,
where 2,1k ; 




 

2
is Jakobi
symbol and      
2 2
1
8
2†
1
; 1 1
2
k y
k
n x y
x
n f

 
  
.
Theorem 5.
Let un 
752 ,   110, u ,
2 2
3 35f x y  ,
22
4 924 yxyxf  ,
2 2
5 4 2 9f x xy y  
Then
     













1
5
11
6
1
;
u
fnr k

    




 











u
kfn
u




;
35
7
1
For 0 ,
   













1
5
11
2
1 u
   




 











u
u



35
7
1 for 2 , 0 ,
0 for 2†,
Where 5,4,3k ; 





5
u
, 





7
u
, 




 
v
35
are
Jacobi symbols and
   
2 2
3
9
2†
2
; 1
3
y
n x xy y
x
n f
  
 
     
2 2
4 5
9
2†
1
; ; 1
3
y
n x xy y
x
n f v n f
  
    .
Theorem 5.
Let un 
752 ,   110, u ,
2 2
6 5 7f x y 
,
22
7 1223 yxyxf  ,
22
8 1223 yxyxf  . Then
     













1
5
11
6
1
;
u
fnr k

    




 











u
kfnv
v
u


;
35
7
1
for 0 ,
   













1
5
11
2
1 u

3. Teimuraz Vep¬khvadze Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 1) January 2016, pp.81-83
www.ijera.com 83|P a g e
   




 











u v
u

 35
7
1 for 2 , 0
0 for 2†,
where 8,7,6k ; 





5
u
, 





7
u
, 




 
v
35
are
Jacobi symbols and
   
2 2
6
3 3
2†
2
; 1
3
x
n x xy y
y
n f
  
   ,
     
2 2
7 8
3 3
2†
1
; ; 1
3
x
n x xy y
y
n f v n f
  
   .
References
[1.] H. Peterson, Modulfunktionen and
quadratischen formen. Ergebnisse der
Mathematik and ihrer Grenzgebeite [Results
in Mathematics and Related Areas], 100.
Springer-Verlag, Berlin, 1982, 71-75.
[2.] P.Kaplan and K.S. Williams, On the number
of representations of a positive integer n by
a binary quadratic form. Acta Arithmetica,
114, 1, 2004, 87-98.
[3.] Teimuraz Vepkhvadze, On the number of
representations of a positive integer by
certain binary quadratic forms belonging to
multi-class genera. International journal of
engineering research and applications
(Ijera), Vol: 5, Issue: 6, 2015, 01-04.
[4.] C.L. Siegel, uber die analytische theorie der
quadratischen formen: Annals of
mathematics, 36, 1935, 527-606.
[5.] T.Vepkhvadze, Modular properties of theta-
functions and representation of numbers by
positive quadratic forms. Georgian Math.
Journal. Vol:4, № 4, 1997, 385-400.