Proofs of some conjectures of Sun on the relations between t(a, b, c; n) and N(a, b, c; n)
Let Z and Z + be the set of integers and the set of positive integers, respectively. For a , b , c , n ∈ Z + and x , y , z ∈ Z , let N ( a , b , c ; n ) be the number of representations of n as a x 2 + b y 2 + c z 2 and t ( a , b , c ; n ) be the number of representations of n as a x ( x + 1 )...
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| Veröffentlicht in: | Indian journal of pure and applied mathematics 2023-12, Vol.54 (4), p.1081-1098 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
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| Zusammenfassung: | Let
Z
and
Z
+
be the set of integers and the set of positive integers, respectively. For
a
,
b
,
c
,
n
∈
Z
+
and
x
,
y
,
z
∈
Z
, let
N
(
a
,
b
,
c
;
n
) be the number of representations of
n
as
a
x
2
+
b
y
2
+
c
z
2
and
t
(
a
,
b
,
c
;
n
) be the number of representations of
n
as
a
x
(
x
+
1
)
2
+
b
y
(
y
+
1
)
2
+
c
z
(
z
+
1
)
2
. Recently, Sun established many relations between
t
(
a
,
b
,
c
;
n
) and
N
(
a
,
b
,
c
;
8
n
+
a
+
b
+
c
)
and listed 43 relations need to be confirmed. More recently, Xia and Zhang proved 19 relations conjectured by Sun. In this paper, by employing Ramanujan’s theta function identities, we prove the remaining 24 relations. |
|---|---|
| ISSN: | 0019-5588 0975-7465 |
| DOI: | 10.1007/s13226-022-00324-8 |