Content uploaded by David Brink

Author content

All content in this area was uploaded by David Brink on Oct 30, 2015

Content may be subject to copyright.

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 + bζ + cζ

2

+ dζ

3

, i.e. p = a

4

+ b

4

+ c

4

+ d

4

+ 2a

2

c

2

+ 2b

2

d

2

+

4a

2

bd−4ab

2

c−4bc

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 + bα + cα

2

+ dα

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 K¨orpers P (

√

− p) mit einer Primzahl p ≡ 1

mod. 2

3

, Aequationes Math. 3 (1969), 165–169.

[7] H. Hasse,

¨

Uber die Klassenzahl des K¨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