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# Two theorems of Glaisher and Kaplansky

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## Abstract

We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher.
Two theorems of Glaisher and Kaplansky
David Brink
October 2008
Abstract. We give a new proof of a recent theorem of Kaplansky and use it
to revive an old, seemingly forgotten result of Glaisher.
It is well known that, for any n > 0, the prime numbers of the form x
2
+ ny
2
can be
described by congruence conditions if and only if n is one of Euler’s convenient numbers
[4, p. 62]. Since 1, 2, 4, 8 and 16 are the only convenient powers of 2, the following
theorem of Kaplansky [8] is remarkable: A prime p 1 (mod 16) is representable by
both or none of x
2
+32y
2
and x
2
+64y
2
, whereas a prime p 9 (mod 16) is representable
by exactly one of these forms. Kaplansky writes, “Although this is a simple elementary
statement I do not have a direct proof. Instead I shall show that it is a quick corollary of
two signiﬁcant theorems.” The latter are reciprocity laws concerning 2 and –4 as fourth
and eighth power residues. In this note we give a new proof of Kaplansky’s theorem
that aspires to be direct. Instead of reciprocity it uses an idea of Aigner [1], Barrucand
and Cohn [2], namely that both representations p = u
2
+ v
2
and p = z
2
+ 2w
2
of a prime
p 1 (mod 8) come from a single representation of p by the norm form of the eighth
cyclotomic ﬁeld.
Consider an odd prime p and let h and h
0
be the class numbers corresponding to the
discriminants 4p and 8p, respectively. Using Kaplansky’s theorem, we give a quick
proof of an old, seemingly forgotten result of Glaisher
1
: If p 1 (mod 16), then either
both or none of h and h
0
are divisible by 8; if p 9 (mod 16), then exactly one of these
class numbers is divisible by 8. This was originally demonstrated in [5, §12] along with
other interesting class number relations using Dirichlet’s class number formula.
Proof of Kaplansky’s theorem. Consider a prime p 1 (mod 8). It splits in the eighth
cyclotomic ﬁeld Q(ζ), ζ
4
+ 1 = 0. Therefore, and since the ring of integers Z[ζ] is a PID,
p is the norm of an integer a + +
2
+
3
, i.e. p = a
4
+ b
4
+ c
4
+ d
4
+ 2a
2
c
2
+ 2b
2
d
2
+
4a
2
bd4ab
2
c4bc
2
d+4acd
2
with a, b, c, d Z. Now u = a
2
c
2
+2bd, v = d
2
b
2
+2ac,
z = a
2
b
2
+ c
2
d
2
and w = ab + cd + ad bc satisfy the identities
p = u
2
+ v
2
= z
2
+ 2w
2
()
1
James W. L. Glaisher (1848–1928), son of the meteorologist and world record holding balloonist of
the same name.
1
where u may be assumed odd. Then v 0 (mod 4), w 0 (mod 2), and u, v, z, w are
all unique modulo sign. One sees immediately from () that the conditions
(1a) u ±1 (mod 8)
(1b) z + 2w ±1 (mod 8)
(1c) p 1 (mod 16)
are equivalent. Since u is odd, a and c must have diﬀerent parity. Hence it follows from
v + z = (a + c)
2
2b
2
±1 (mod 8) that
(2a) v 0 (mod 8)
(2b) z ±1 (mod 8)
are equivalent. Combining this with the above gives that also
(3a) u + v ±1 (mod 8)
(3b) w 0 (mod 4)
are equivalent, a fact also contained in [2, Main Theorem]. Finally, it is clear that either
all three or only one of (1a), (2a) and (3a) holds. Consequently, either all three or only
one of (1c), (2a) and (3b) holds, which concludes the proof.
Proof of Glaisher’s theorem. It is an immediate consequence of the Gaussian theory of
genera that 2 | h if and only if p 1 (mod 4), and that always 2 | h
0
. Glaisher showed
that 4 | h if and only if p 1 (mod 8), and that 4 | h
0
if and only if p ±1 (mod 8).
Furthermore, 8 | h if and only if p is of the form x
2
+ 32y
2
, and 8 | h
0
if and only if p is
either of the form x
2
+64y
2
or 1 (mod 16). The ﬁrst of these two beautiful theorems
was proved by Barrucand and Cohn and later, in a diﬀerent manner, by Hasse [6]. The
second was proved for p 1 (mod 8) by Glaisher and for all p by Hasse [7]
2
. Glaisher’s
and Kaplansky’s theorems are now seen to follow from one another.
Five results similar to Kaplansky’s theorem were found in [3], for example the following:
A prime p 1 (mod 20) is representable by both or none of x
2
+ 20y
2
and x
2
+ 100y
2
,
whereas a prime p 9 (mod 20) is representable by exactly one of these forms. The
proof used class ﬁeld theory. As a ﬁnal remark, we give here a diﬀerent, more elementary
demonstration using the same basic idea as above: Consider a prime p 1, 9 (mod 20).
It splits in the ﬁeld Q(
5, i) = Q(α) where α
4
+ 3α
2
+ 1 = 0. Since the ring of integers
Z[α] is a PID, p is the norm of an integer a + +
2
+
3
, i.e. p = a
4
+ b
4
+
c
4
+ d
4
+ 3a
2
b
2
+ 11a
2
c
2
+ 18a
2
d
2
+ 3b
2
c
2
+ 11b
2
d
2
+ 3c
2
d
2
6a
3
c 6ac
3
6b
3
d
6bd
3
14a
2
bd 4ab
2
c 14acd
2
4bc
2
d + 12abcd. Hence p = u
2
+ v
2
= z
2
+ 5w
2
with
u = a
2
+ b
2
+ c
2
+ d
2
3ac 3bd, v = 4ad ab bc cd, z = a
2
b
2
+ c
2
d
2
3ac + 3bd
and w = 2ad + ab bc + cd. The statement can now be shown as above, but also
2
But note that the relevant form erroneously appears to be x
2
+16y
2
. In fact, Hasse proves a diﬀerent
criterion for primes p 1 (mod 8) and refers to a private communication from Barrucand in which this
was shown to be equivalent to p = x
2
+ 64y
2
. Hasse seems to have been unaware of Glaisher’s results
and also refers to much later works of R´edei and Reichardt regarding the criterion for 4 | h
0
.
2
by brute force simply by letting a, b, c, d run through all residue classes modulo 20 and
checking the assertion in each case.
References
[1] A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und –2, Deutsch.
Math. 4 (1939), 44–52.
[2] P. Barrucand, H. Cohn, Note on primes of type x
2
+ 32y
2
, class number, and resid-
uacity, J. Reine Angew. Math. 238 (1969), 67–70.
[3] D. Brink, Five peculiar theorems on simultaneous representation of primes by
quadratic forms, J. Number Theory 129 (2009), 464–468.
[4] D. A. Cox, Primes of the form x
2
+ ny
2
, Wiley, New York, 1989.
[5] J. W. L. Glaisher, On the expressions for the number of classes of a negative deter-
minant, and on the numbers of positives in the octants of P , Quart. J. Pure Appl.
Math. 34 (1903), 178–204.
[6] H. Hasse,
¨
Uber die Klassenzahl des orpers P (
p) mit einer Primzahl p 1
mod. 2
3
, Aequationes Math. 3 (1969), 165–169.
[7] H. Hasse,
¨
Uber die Klassenzahl des orpers P (
2p) mit einer Primzahl p 6= 2, J.
Number Theory 1 (1969), 231–234.
[8] I. Kaplansky, The forms x + 32y
2
and x + 64y
2
, Proc. Amer. Math. Soc. 131 (2003),
no. 7, 2299–2300 (electronic).
3
... The formulas in Table 2.3 imply these results as follows: (1,1,1 , as shown below. According to Brink (2009) the property (P1) even characterizes the primes ) 4 (mod 1 ≡ p , a result already derived by Glaisher (1903) (see also Lerch (1906), p. 224). Glaisher also characterizes the set of all odd primes by the property (P2). ...
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For the specific set of fifteen ternary quadratic forms x^2+by^2+cz^2, b,c in {1,2,4,8}, (b,c) in {(2,16),(8,16),(1,3),(2,3),(1,5)} it is shown that the distinct zero-free representations of an odd prime by these forms depend upon the class numbers h(-kp), k in {1,3,4,5,8,12,20,24}. We determine when such a form is universal zero-free for an arithmetic progression of primes, i.e., when a prime from such a progression can be represented without zero components. The exceptional primes, which cannot be represented in this way, fall into two distinct classes. They are either infinite in number and belong to arithmetic progressions of primes, so-called infinite exceptional sets, or they are finite in number and build so-called finite exceptional sets. These exceptional sets are determined. Moreover, we show how to derive the finite number of primes expressible by a form x^2+by^2+cz^2 in essentially m ways, and illustrate the method. Reinterpreting results by Dirichlet (1850), Dickson (1927) and Kaplansky (1995), we show that the forms (b,c) in {(1,2),(1,3),(2,3),(2,4)} are the only strictly universal zero-free forms of type x^2+by^2+cz^2, i.e., they can be represented without zero components for all primes up to a known finite number of primes.
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It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x² + 32y² and x² + 64y², whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved, and a heuristic argument is given why there are no other results of the same kind. The latter argument relies on the (plausible) conjecture that the list of 485 known negative discriminants Δ such that the class group C(Δ) has exponent 4 is complete. The methods are purely classical.
The forms x + 32y 2 and x + 64y 2
• I Kaplansky
I. Kaplansky, The forms x + 32y 2 and x + 64y 2, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2299-2300 (electronic).
Kriterien zum 8. und 16
• A Aigner
A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und –2, Deutsch. Math. 4 (1939), 44–52.
On the expressions for the number of classes of a negative determinant , and on the numbers of positives in the octants of P , Quart
• J W L Glaisher
J. W. L. Glaisher, On the expressions for the number of classes of a negative determinant, and on the numbers of positives in the octants of P, Quart. J. Pure Appl. Math. 34 (1903), 178–204.