80. ON PARTITIONS INTO FOUR
DISTINCT SQUARES OF EQUAL PARITY
1. Introduction and statement of results
In recent work [1,2], we studied p4 (n), the number of partitions of n into four squares.
We found the generating function for p4 (n), and showed that the numbers p4 (n) possess
some (rare) arithmetic properties.
In this paper we make a study of what we will call p4e (n) and p4o (n), the number of
partitions of n into four distinct even (respectively odd) squares. We shall also have
reason to consider p4e+ (n), where the squares are strictly positive.
The main result of this paper is that for n ≡ 4 (mod 8),
p4o (n) = p4e (n) + p4e+ (n).
As a corollary we obtain the surprising result that for n ≡ 4 (mod 8),
p4o (n) ≥ p4e (n) ≥
1
p4o (n).
2
Thus, if p4o (n) > 0 (and so n ≡ 4 (mod 8)) then p4e (n) > 0, a conjecture announced by
R. Wm. Gosper at the Ramanujan Centenary Conference in 1987, the inspiration for this
investigation.
We shall give two proofs of our theorem, one arithmetic, the other via generating
functions.
2. The arithmetic proof
We shall show that if n ≡ 4 (mod 8) there are two partitions of n into four distinct odd
squares for each partition of n into four distinct even squares greater than zero, and one
partition of n into four distinct odd squares for each partition of n into four distinct even
squares one of which is zero.
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1
2
80. ON PARTITIONS INTO FOUR DISTINCT SQUARES OF EQUAL PARITY
Suppose n ≡ 4 (mod 8) and
n = x21 + x22 + x23 + x24 ,
is a partition of n into four distinct even squares. Then w.l.o.g.
(x1 , x2 , x3 , x4 ) ≡ (0, 2, 2, 2) or (2, 0, 0, 0) (mod 4),
x1 + x2 + x3 + x4 ≡ 2
(mod 8)
and
x2 > x3 > |x4 |.
(In the case (x1 , x2 , x3 , x4 ) ≡ (0, 2, 2, 2) (mod 4), the condition x1 + x2 + x3 + x4 ≡
2 (mod 8) determines the sign on x4 , (the sign of x1 is undetermined); in the case
(x1 , x2 , x3 , x4 ) ≡ (2, 0, 0, 0) (mod 4) the condition determines the sign on x1 (the sign
of x4 is undetermined).)
In the case (x1 , x2 , x3 , x4 ) ≡ (0, 2, 2, 2) (mod 4), let
 1
y1
2
1
 y2  

  =  12
y3
2
1
y4
2

1
2
1
2
− 21
− 21
1
2
− 21
1
2
− 21
1
2
− 21
− 21
1
2
  ′ 1
|x1 |
y1
2
  x2   y2′   1
2


 x3  ,  y3′  =  1
2
1
x4
y4′
2

1
2
1
2
− 12
− 12
1
2
− 12
1
2
− 12
1
2
− 21
− 21
1
2


−|x1 |
  x2 

.

x3
x4
Then
2
2
2
2
n = y12 + y22 + y32 + y42 = y1′ + y2′ + y3′ + y4′ ,
y1 , y2 , y3 , y4 ≡ 1 (mod 4), y1′ , y2′ , y3′ , y4′ ≡ 1 (mod 4),
y1 > y2 > y3 > y4 , y1′ > y2′ > y3′ > y4′
are partitions of n into four distinct odd squares, different if and only if x1 6= 0.
Similarly, in the case (x1 , x2 , x3 , x4 ) ≡ (2, 0, 0, 0) (mod 4), let
 1
y1
2
1
 y2  

  =  12
y3
2
1
y4
2

1
2
1
2
− 21
− 21
1
2
− 21
1
2
− 21
1
2
− 21
− 21
1
2
  ′ 1
x1
y1
2
  x2   y2′   1
2


 x3  ,  y3′  =  1
2
1
|x4 |
y4′
2

1
2
1
2
− 12
− 12
1
2
− 12
1
2
− 12
Then
2
2
2
2
n = y12 + y22 + y32 + y42 = y1′ + y2′ + y3′ + y4′ ,
1
2
− 21
− 21
1
2


x1
  x2 

.

x3
−|x4 |
80. ON PARTITIONS INTO FOUR DISTINCT SQUARES OF EQUAL PARITY
3
y1 , y2 , y3 , y4 ≡ 1 (mod 4), y1′ , y2′ , y3′ , y4′ ≡ 1 (mod 4),
y1 > y2 > y3 > y4 , y1′ > y2′ > y3′ > y4′
are partitions of n into four distinct odd squares, different if and only if x4 6= 0.
Conversely, suppose n ≡ 4 (mod 8) and
n = y12 + y22 + y32 + y42
is a partition of n into four odd squares. Then w.l.o.g.
y1 , y2 , y3 , y4 ≡ 1
If we now let
 1
x1
2
1
 x2  

  =  21
x3
2
1
x4
2

then
(mod 4) and y1 > y2 > y3 > y4 .
1
2
1
2
− 12
− 12
1
2
− 12
1
2
− 12
1
2
− 21
− 21
1
2


y1
  y2 
 
 y3
y4
n = x21 + x22 + x23 + x24 ,
(x1 , x2 , x3 , x4 ) ≡ (0, 2, 2, 2) or (2, 0, 0, 0) (mod 4),
x1 + x2 + x3 + x4 ≡ 2
(mod 8)
and
x2 > x3 > |x4 |.
This establishes the (one–to–one–or–two) correspondence described earlier between the
partitions of n into four distinct even squares and the partitions of n into four distinct
odd squares. An example
Consider n = 420. The partitions of 420 into four distinct even squares are
420 = 202 + 42 + 22 = 162 + 122 + 42 + 22 = 162 + 102 + 82 = 142 + 122 + 82 + 42 .
Corresponding to these we have, respectively,
     

2
13
x1
y1
 x2   20   y2   9 
,
  =  ,   = 
4
−7
x3
y3
x4
y4
0
−11

4
80. ON PARTITIONS INTO FOUR DISTINCT SQUARES OF EQUAL PARITY
 
   
  ′ 

x1
2
y1
17
y1
13
′
 x2   16   y2   1   y2   5 
 =
,   = 
,  ′  = 
,
x3
12
y3
−3
y3
1
′
x4
±4
y4
−11
y4
−15

     

x1
10
y1
17
 x2   16   y2   9 
  =  ,   = 
,
x3
8
y3
1
x4
0
y4
−7

 
   
  ′ 

x1
−14
y1
5
y1
1
′
 x2   12   y2   −7   y2   −3 
 =
,   = 
,  ′  = 
.
8
x3
−11
y3
y3
−7
′
x4
±4
y4
−15
y4
−19

Thus
420 = 202 + 42 + 22 = 162 + 122 + 42 + 22 = 162 + 102 + 82 = 142 + 122 + 82 + 42
= 192 + 72 + 32 + 12 = 172 + 112 + 32 + 12 = 172 + 92 + 72 + 12
= 152 + 132 + 52 + 12 = 152 + 112 + 72 + 52 = 132 + 112 + 92 + 72 ,
p4o (420) = 6, p4e (420) = 4, p4e+ (420) = 2, and p4o (420) = p4e (420) + p4e+ (420).
3. The generating function proof
Let xi , i = 0, · · · , N be arbitrary. It is easy to verify by expanding the expression on
the right that
(1)
24
X
xi xj xk xl
i>j>k>l
=
N
X
i=0
xi
!4
N
X
−6
i=0
xi
!2
N
X
x2i
+3
i=0
N
X
x2i
i=0
!2
+8
N
X
i=0
xi
N
X
i=0
x3i − 6
N
X
x4i .
i=0
[The author discovered this formula recently, but feels that it is unlikely to be new.]
2
If in (1) we set xi = q (2i+1) , let N → ∞ and use the fact that
X
i≥0
2
q (2i+1) = qψ(q 8 )
80. ON PARTITIONS INTO FOUR DISTINCT SQUARES OF EQUAL PARITY
5
where
ψ(q) =
X
2
q (i
+i)/2
,
i≥0
we obtain
(2)
X
p4o (n)q n =
n≥0
1 4
q ψ(q 8 )4 − 6q 4 ψ(q 8 )2 ψ(q 16 ) + 3q 4 ψ(q 16 )2 + 8q 4 ψ(q 8 )ψ(q 24 ) − 6q 4 ψ(q 32 )
24
or,
X
p4o (n)q n =
n≡4 (mod 8)
1 4
q ψ(q 8 )4 − 6q 4 ψ(q 8 )2 ψ(q 16 ) + 3q 4 ψ(q 16 )2 + 8q 4 ψ(q 8 )ψ(q 24 ) − 6q 4 ψ(q 32 )
24
since the series in (2) contains only powers of q which are 4 modulo 8.
2
If in (1) we set xi = q (2i) , let N → ∞ and use the fact that
X
2
q (2i) =
i≥0
1
(φ(q 4 ) + 1)
2
where
φ(q) =
∞
X
2
qi ,
−∞
we obtain, after simplification,
(3)
X
n≥0
p4e (n)q n =
1
φ(q 4 )4 + 4φ(q 4 )3 − 6φ(q 4 )2 + 12φ(q 4 ) − 15 + 12φ(q 8 )2 + 12φ(q 8 )
384
+32φ(q 12 ) − 48φ(q 16 ) − 12φ(q 4 )2 φ(q 8 ) − 24φ(q 4 )φ(q 8 ) + 32φ(q 4 )φ(q 12 ) .
2
If in (1) we set xi = q (2i+2) , let N → ∞ and use the fact that
X
i≥0
2
q (2i+2) =
1
(φ(q 4 ) − 1),
2
6
80. ON PARTITIONS INTO FOUR DISTINCT SQUARES OF EQUAL PARITY
we obtain
(4)
X
p4e+ (n)q n =
n≥0
1
φ(q 4 )4 − 4φ(q 4 )3 + 18φ(q 4 )2 − 60φ(q 4 ) + 105 + 12φ(q 8 )2 − 36φ(q 8 )
384
−32φ(q 12 ) − 48φ(q 16 ) − 12φ(q 4 )2 φ(q 8 ) + 24φ(q 4 )φ(q 8 ) + 32φ(q 4 )φ(q 12 ) .
It follows from (3) and (4) that
X
1
φ(q 4 )4 + 6φ(q 4 )2 − 24φ(q 4 ) + 45 + 12φ(q 8 )2 − 12φ(q 8 )
192
−48φ(q 16 ) − 12φ(q 4 )2 φ(q 8 ) + 32φ(q 4 )φ(q 12 ) .
(p4e (n) + p4e+ (n))q n =
n≥0
We now extract the terms in which n ≡ 4 (mod 8). We use the fact that
φ(q) = φ(q 4 ) + 2qψ(q 8 )
to write
X
(p4e (n) + p4e+ (n))q n =
n≥0
1
(φ(q 16 ) + 2q 4 ψ(q 32 ))4 + 6(φ(q 16 ) + 2q 4 ψ(q 32 ))2
192
−24(φ(q 16 ) + 2q 4 ψ(q 32 )) + 45 + 12φ(q 8 )2 − 12φ(q 8 ) − 48φ(q 16 )
−12(φ(q 16 ) + 2q 4 ψ(q 32 ))2 φ(q 8 )
+32(φ(q 16 ) + 2q 4 ψ(q 32 ))(φ(q 48 ) + 2q 12 ψ(q 96 )) .
It follows that
X
(p4e (n) + p4e+ (n))q n
n≡4 (mod 8)
=
=
=
1 4
q φ(q 16 )3 ψ(q 32 ) + 4q 12 φ(q 16 )ψ(q 32 )3 + 3q 4 φ(q 16 )ψ(q 32 ) − 6q 4 ψ(q 32 )
24
−6q 4 φ(q 8 )φ(q 16 )ψ(q 32 ) + 8q 4 φ(q 48 )ψ(q 32 ) + 8q 12 φ(q 16 )ψ(q 96 )
1 4
q φ(q 8 )2 φ(q 16 )ψ(q 32 ) − 6q 4 φ(q 8 )φ(q 16 )ψ(q 32 ) + 3q 4 φ(q 16 )ψ(q 32 )
24
+8q 4 ψ(q 8 )ψ(q 24 ) − 6q 4 ψ(q 32 ) .
1
ψ(q 4 )4 − 6q 4 ψ(q 8 )2 ψ(q 16 ) + 3q 4 ψ(q 16 )2 + 8q 4 ψ(q 8 )ψ(q 24 ) − 6q 4 ψ(q 32 ) .
24
80. ON PARTITIONS INTO FOUR DISTINCT SQUARES OF EQUAL PARITY
7
where we have used the facts that
φ(q 2 )2 + 4qψ(q 4 )2 = φ(q)2 ,
φ(q 6 )ψ(q 4 ) + qφ(q 2 )ψ(q 12 ) = ψ(q)ψ(q 3 )
and
φ(q)ψ(q 2 ) = ψ(q)2 . References
[1] M. D. Hirschhorn, Some formulae for partitions into squares, Discrete Math, 211(2000),225–
228.
[2] M. D. Hirschhorn and J. A. Sellers, Some relations for partitions into four squares, in
Special Functions 1999, Proceedings of the International Workshop on Special Functions,
Hong Kong, World Scientific.