extensions of theorems of cunningham-aigner and hasse-evans

Pacific Journal ofMathematics

EXTENSIONS OF THEOREMS OF CUNNINGHAM-AIGNERAND HASSE-EVANS

RICHARD HOWARD HUDSON AND KENNETH S. WILLIAMS

Vol. 104, No. 1 May 1983

PACIFIC JOURNAL OF MATHEMATICSVol. 104, No. 1, 1983

EXTENSIONS OF THEOREMS OFCUNNINGHAM-AIGNER AND HASSE-EVANS

RICHARD H. HUDSON AND KENNETH S. WILLIAMS

If £ is a positive integer andp is a prime with/? = 1 (mod2k), then2(/>-i)/2 j s a 2*th root of unity modulo p. We consider the problemof determining 2 ( / ? ~ 1 ) / 2 * modulo p. This has been done for k - 1, 2, 3and the present paper treats k = 4 and 5, extending the work ofCunningham, Aigner, Hasse, and Evans.

1. Introduction. When k — 1, we have the familiar result

{ - 1 (mod /?), ifp = 3, 5 (mod 8).

When k — 2 and p = 1 (mod 4), there are integers 0 = 1 (mod 4) andb = 0 (mod 2) such that /? = α2 + Z?2, with a and | Z? | unique. If b = 0(mod 4) (so that /? = 1 (mod 8)), Gauss [8: p. 89] (see also [4], [16]) hasshown that

(1 2Ϊ Ύ i f δ Ξ

K ' ) \ - 1 (mod/?), if 6 ΞΞ 4 (mod 8).

If Z> = 2 (mod 4) (so that p = 5 (mod 8)), we can choose b = -2 (mod 8),by changing the sign of 6, if necessary, and Gauss [8: p. 89] (see also [4],[11: p. 66], [16]) has shown that

(1.3) 2{p~ι)/4 Ξ -b/a (mod/?).

We note that (-b/a)2 = -1 (mod p).When k = 3 and /? = 1 (mod 8), there are integers a = 1 (mod 4) and

6 = 0 (mod 4) such that p — a2 + b2, with a and | b \ unique. Now{2^-D/8j4 = 2(^-D/2 = ! (mod/?), as/? = 1 (mod 8), so 2^-])^s is a 4throot of unity modulo p. If b = 0 (mod 8), Reuschle [14] conjectured andWestern [15] (see also [16]) proved that

(14) 2 ( , - . ) / 8 Ξ ί ( - 1 ) ( / " 1 ) / 8 ( m o d / ' ) ' if* = 0 (mod 16),

$-lfp+Ί)/s(modp), ifZ>Ξ=8(modl6).

If b = 4 (mod 8), we can choose b = 4(-$(p+1)/s (mod 16), by changingthe sign of b, if necessary, and Lehmer [11: p. 70] has shown that

(1.5) 2<"-'>/8 = --a (mod/)).

I l l

112 RICHARD H. HUDSON AND KENNETH S. WILLIAMS

It is the purpose of this paper to treat the cases k — 4 and 5. Fork — 4 and p = 1 (mod 16), there are integers a = 1 (mod 4), b = 0(mod 4), c = 1 (mod 4), d = Q (mod 2), such that p - a2 Λ- b2 - c2

+ 2d2, with a, |Z>|, c, |έ/| unique. Now { ^ " ^ f r z(mod/?), so 2 ( ; 7~ 1 ) / 1 6 is an 8th root of unity modulo/?. Since

(1.6)

the 8th roots of unity (mod p) are given by {— (a + b)d/ac}n, n —0,1, . . . ,7. Making use of a congruence due to Hasse [9: p. 232] (see also[5: Theorem 3], [17: p. 411]), we prove in §2 the following extension of thecriterion for 2 to be a 16th power (mod /?), which was conjectured byCunningham [3: p. 88] and first proved by Aigner [1] (see also [16:p. 373]).

THEOREM 1. Let p Ξ 1 (mod 16) be a prime. Let a = 1 (mod 4 ) , ί ) Ξ θ(mod 4), c = 1 (mod 4), d = 0 (mod 2) be integers such that p — a2 + b2

= c2 + Id2. It is well known that b = 0 (mod 8) >̂ d = 0 (mod 4) (see forexample [2: p. 68]). Then the values of 2(p~ι)/ιβ (mod p) are given inTable 1.

The case b = 0 (mod 16) constitutes the criterion of Cunningham-Aigner.

For k — 5 and p = 1 (mod 32), there are integers a = 1 (mod 4),b = 0 (mod 4), c ΞΞ 1 (mod 4), d = 0 (mod 2), x = -1 (mod 8), w = ϋ Ξ=

w = 0 (mod 2), such that/? = a2 + b2 = c2 + Id2 and

( ι 7) Ϊp = x2 + 2u2 + 2v2 + 2w2,

[2 xv = u2 — 2ww — w 2 ,

with a, IZ? I , c, | J | , x unique. If (x, w, ϋ, w) is a solution of (1.7), then allsolutions are given by ± ( x , w, υ, w), =t(x, —w, v, —w), ±(x9 w, —v, — u),± ( J C , - W , - o , w) (see for example [12: p. 366]). Now {2^"1>/3 2}1 6 =2</>-υ/2 = + 1 ( m o d p^ s o 2(P- ] )/32 i s a i 6 t h r o o t o f u n i t y modulo /?.

Since

I ( A + „ ) ( , ( • • + » ) * ( • . » • ) ) 1 E ( « + t V ( m o d )

1 2 W ( « + w ) J α c

the 16th roots of unity (mod/?) are given by

w)-b(u-w))Y

EXTENSIONS OF THEOREMS OF CUNNINGHAM-AIGNER 113

Making use of another congruence due to Hasse [9: p. 233] (see also [7:eqn. (2)]), we prove in §3 the following extension of the criterion for 2 tobe a 32nd power (mod /?) due to Hasse [9: p. 232-238] and Evans [6:Theorem 7].

THEOREM l.Letp = 1 (mod 32) be a prime. Let a = 1 (mod 4), b = 0(mod 4), c ΞΞ 1 (mod 4), d = Q (mod 2), x = -1 (mod 8),u = v = w = 0(mod 2), be integers such that p — a2 + b2 = c2 + Id2 and p — x2 +2u2 + 2v2 + 2w2, 2xv = u2 - 2uw - w2. Then the values 2^~ 1 ) / 3 2

(mod/?) are g/t)έ?i IΛ Table 2.

Justification of the choices in the left-hand column of Table 2 is madein the proof of Theorem 2, which appears in §3. The cases 2{p~λ)/32 =±l(mod p) constitute the criterion of Hasse-Evans.

2. Evaluation of 2{p~1)/16 (modp). Letp be a prime satisfying

(2.1) p = 1 (mod 16).

Set

(2.2) /> = 8 / + l ,

so that(2.3) /Ξ=0(mod2) .

Let

(2.4) ω = exp(2τr//8) = (1 + i)/fi.

We note that the ring of integers of Q(ω) — Q(i, ]/2) is a unique factori-zation domain (see for example [13]). In this ring/? factors as a product offour primes. Denoting one of these by π, these four primes are 7ry = σ (flr),j — 1, 3, 5, 7, where Oj denotes the automorphism which maps ω to ωJ.

Let g be a primitive root (mod/?). Then g ^ " 1 ) / 2 = -1 (mod/?), andso

U 7 ~ ω ) ( g 7 - <*3)(gf - ω 5 )(g / - ω7) = 0 (mod ΊT^TT^).

Hence

gf — ωJ = 0 (mod π,),

for some j , j = 1, 3, 5, 7, and by relabelling the TΓ'S we may assumewithout loss of generality that

(2.5) gf = ω (mod TΓ).

Given g, TΓ (apart from units) is uniquely determined by (2.5). Next wedefine a character χ (mod/?) (depending upon g) of order 8 by setting

(2.6) χ(g) = ω.


For r, s = 0, 1, 2,... ,7 the Jacobi sum/(r, s) is defined by

(2.7) J(r,s)= Σ Xr(n)χs(l-n).n (mod/?)

It is known that (see for example [7: §1])

(2.8) J(2,2) = -a + bi,

where

(2.9) p = a2 + b2, a = \ (mod 4),

and that

(2.10) /(1,3) = -c + ώj2,

where

(2.11) p = c2 + 2d2, c= 1 (mod 4).

It is easy to check that replacing the primitive root g by the primitive rootg 8 ί + ' , where f = 1, 3, 5, 7 and (8* + / , / ) = 1, has the effect in (2.8) ofreplacing b by(—l/t)b and in (2.10) of replacing d by (~2/t)d.

Our proof depends upon the following important congruence due toHasse [9: p. 232]

(2.12) b = 4d+2m(mod32),

where m is the least positive integer such that

(2.13) g" = 2(

and b and d are given by (2.8) and (2.10) respectively. From (2.12) and(2.13) we obtain

(2.14) 2<'~1 )/1 6 = 2"2 = g*'/ 2 = g/<*/4-«o (mod/;).

It follows from (2.5) and (2.6) that

(2.15) χ(n) ΞΞH^modTΓ),

for any integer n not divisible by p. Hence, for non-negative integers r ands satisfying 0 < r + s < 8, we have

P-\J(r,s)= £ nr/(l-n)sf(modπ)

n = 0

sf

= 2 "rfΣ (-l)V(modτr)Λ2 = 0 y = 0 \ J I

sf

Σj=0


that is

(2.16) J(r9s) Ξθ(modτr),

as

p-\

(2.17) 2 nk = 0 (mod/?), for k = 0, 1,...,/?- 2.

Taking (r, J ) = (2,2) and (1,3) in (2.16), we have, by (2.8) and (2.10),

(2.18) i = a/b (mod 77), 1/2 Ξ c/d (mod TΓ),

so that

(2.19) /2 = -ac/bd (mod TΓ).

Hence we have, appealing to (2.5), (2.18) and (2.19),

f 1 + 1 (a + b)d ( ,

~ΊΓ Ξ "—~™—and, since g^and —(a + b)d/ac are integers (mod/?), we have

(2.20) gf^-{a+J)d (modp).

Appealing to (2.14) we get

We consider three cases:

(ii) 2<p-χ)/4 ΞΞ + 1 , 2 ( '- 1>/ 8 = -1 (mod/?),= + 1 (mod/?).

(i). From (1.2) we have b = 4 (mod 8). Then, from/? = β 2 + 62,we obtain a = 1 (mod 8) and /? Ξ 2a + 15 (mod 32). The cyclotomicnumber (0,7)8 is given by (see for example [10: p. 116])

64(0,7)8 =p - Ί + 2a + 4c,

so c = 5 (mod 8). Then, from p = c2 + 2d2, we get d = 2 (mod 4).Replacing g by an appropriate primitive root


we may take b ~ -4 = 12 (mod 16) and d = 2 (mod 8). Then, from (2.21),we obtain

2(p-\)/l6 Ξ ,

(a + b)d

ac(a + b)d

ac

(mod/?), ifft = 12 (mod 32),

(modp), if b = 28 (mod 32).

(ii). From (1.2) and (1.4) we have b = 8 (mod 16). Then, fromp — a2 + ft2, we obtain α = 1 (mod 8) and p = 2a — 1 (mod 32). Thecyclotomic number (1,2)8 is given by (see for example [10: p. 116])

64(1,2) 8 =p+l+2a- 4c,

so c = 1 (mod 8). Then, from p — c2 + 2d2, we get d = 0 (mod 4).Replacing g by an appropriate primitive root

we may take b = 8 (mod 32). Then as

j Ξ — (mod/?),

we have from (2.21)

2(/,-i)/i6 = ί - f e A

{, if d = 0 (mod 8),

{ +b/a (mod/?), if J = 4 (mod 8).

(iii). From (1.4) we have 6 Ξ O (mod 16). Exactly as in Case (ii)we have d = 0 (mod 4). Considering four cases according as b =0, 16 (mod 32) and d = 0,4 (mod 8) we obtain from (2.21)

+ 1 (mod/?), if ft Ξ 0 (mod 32), d = 0 (mod 8)or

b = 16 (mod 32), d = 4 (mod 8),

-1 (mod/?), if ft = 0 (mod 32), d = 4 (mod 8)or

ft ΞΞ 16 (mod 32), </ = 0 (mod 8).

This completes the proof of Theorem 1.

3. Evaluation of 2 ( p ~ 1 ) / 3 2 (mod/?). Let/? be a prime satisfying

(3.1) /? = 1 (mod 32).


Set

(3.2) / > = 1 6 / + 1 ,

so that

(3.3) / = 0 ( m o d 2 ) .

Let

(3.4) θ = exp(2«/16) =

Again, the ring of integers of <2(0) is a unique factorization domain (seefor example [13]). In this ring p factors as a product of eight primes.Denoting one of these by π, these eight primes are given by tπi = σ,(τ7),i = 1, 3, 5, 7, 9, 11, 13, 15, where σi denotes the automorphism whichmaps θ to θ\

Let g be a primitive root (modp). Then

(gf ~ θ)(gf - θ3) (g/ - 015) = 0 (mod πxπ3 τr15)9

and, as before, we can choose πι — m (unique apart from units) so that

(3.5) gf = θ(modπ).

We define a character Ψ (mod/?) of order 16 by setting

(3.6) Ψ(g) - θ,

and for r, s = 0, 1, 2,..., 15 we define the Jacobi sum J(r, s) by

(3.7) J(r,s)= 2 V(n)Πl ~ «)•n (mod/>)

It is known that (see for example [7: §1])

(3.8) /(4,4) = -a + bi, where/? = α2 + 62, α Ξ 1 (mod 4),

(3.9) J(2,6) = -c + ί/z/2 , where/7 = c2 + 2d2, c = ί (mod 4),

and

(3.10) /(1,7) = JC + i«γ2 - /2 + ϋ/2 + w i ^ + /2

where (see for example [5; eqn. (8)])

(3 11) ί/? = x 2 + 2M2 + 2«2 + 2w2, x ΞΞ-1 (mod 8),

I 2xv = u2 — 2uw — w2.


It is easy to check that w, υ and w are all even. Applying the mapping0->03 to (3.10), we obtain

(3.12) J(3,5) = x - wφ - /2 -vfi+ uφ + {ϊ

= χ-w(θ + θ1) - V{θ2 ~ θβ) + U(θ3 + θ5).

Further, it is known (see [12: p. 366] and [6: eqn. (48)]) that a, b, c, d, x, w,t>, w are related by

(3.13) bd(x2 - 2υ2) == ac(u2 + 2uw - w2) (mod^).

The effect on (3.8), (3.9), (3.10) of replacing the primitive root g by

the primitive root gλβs+\ where / = 1, 3, 5 , . . . , 15 and (16s + / , / ) = 1, is

summarized below:

gg l 6 s + 3

g l 6 ί + 5

g l 6 ί + 7

g . 6 , + 916̂ +11

g . 6 ί + , 3

16.+15

aaaaaaaa

b-bb-bb-bb-b

cccccccc

dd-d-ddd-d-d

X

X

X

X

X

X

X

X

uwwu————

uwwu

V

— υ

— v

V

υ— v

— υ

υ

w— u

— u

w— w

uu— w

(3.14)

The following important congruence relating b, d, u and w has beenproved by Hasse [9: p. 233]

(3.15) b + 4d- 8(M + W) Ξ 2?M (mod 64),

where m satisfies (2.13). From (2.13) and (3.15), we obtain

(3.16) 2 ( ' - 1 ) / 3 2 = 2f/1 = gmf/2•= g/«V4)+rf-2(»+w)) ( m o d / > ) .

As in §2, if r and s are non-negative integers satisfying 0 ^ r + s < 16, wehave

(3.17) / ( r , ί ) Ξ θ ( m o d τ r ) .

Thus, in particular, taking (/% s) = (4,4), (2,6), (1,7), and (3,5), in (3.17),we obtain

(3.18) - a + 6/Ξ

(3.19) -c + dii/ϊ Ξθ(modir),

(3.20) x + uiyjl - fϊ + φ + wiyjl + {ϊ = 0 (mod π),

(3.21) x - wiyjl - /2 - ϋ/2 + ui^l + ft = 0 (mod «r).


From (3.18) and (3.19) we get

(3.22) / == a/b (mod π), i{λ Ξ= c/d (mod π),

{l = - — (mod 77).

Solving (3.20) and (3.21) simultaneously for ^2 + y/ϊ and /2(mod 77), and making use of (3.22), we obtain

, Λ . r: /=" x(u ± w)ad + υ(u + w)bc , Λ λ

(3.23) 1/2 ± /2 Ξ - i J v '— (mod ff).

Then, from (3.4), (3.5), (3.22) and (3.23), we have

(3.24) g / Ξ ^ ( ^ + ^ ) ( ^ + ^ - ^ - w ) )2bd(u2 + w2)

Since both sides of (3.24) are integers (mod/?), we deduce that

(3.25)uz

Appealing to (3.16) we get

(3 26) 2' f { d X

2bd(u2 + w2)

(mod p).

We consider four cases:(1)2^-0/4 = _i(modp),

(ii) 2<p-χyΛ Ξ + 1, 2^-"/ 8 = -1 (mod/?),(iii)2(/'-1)/8 = + i , 2 ( ^ - 1 ) / ' 6 Ξ -1 (mod/>),(iv)2 ( ί >-1 ) / 1 6== + i (mod/?).

(i). From Case (i) of §2 we have b = 4 (mod 8) and d = 2(mod 4). Next, from (2.12) and (3.15), we obtain

u + w Ξ d = 2 (mod 4),

so that

(u,w) =(0,2) or (2,0) (mod 4).

Replacing g by an appropriate primitive root gl6s+t (where t = 1, 3, 5,...,15 and (16s + ί, f) = 1), we can suppose that

(3.27) b = -4 (mod 16), w HE 0 (mod 4), w s 2 ( m o d 8 ) .


Exactly one 5-tuple (b, d, w, v, w) satisfies (3.13) and (3.27). Then, from2xυ — u2 — 2uw — w 2, we obtain (recalling x = - 1 (mod 8))

(3.28) ϋ = 2 (mod 8).

F r o m the work of Evans and Hill [7: Table 2a], we have

(3.29) 256{(2,4) 1 6 - (4,10) 1 6 } - 32(ι> - d),

so that, by (3.28),

(3.30) d = υ = 2 (mod 8).

The choice (3.27) makes the exponent (b/4) + d~ 2(u + w) in (3.26)congruent to 1 (mod 4). We now consider cases according as b = 12, 28,44, 60 (mod 64); d = 2, 10 (mod 16); M Ξ O , 4 (mod 8). F o r example, ifb ΞΞ 12 (mod 64), d = 2 (mod 16), u = 0 (mod 8), then (b/4) + d -2(u + w) = 1 (mod 16), so that (3.26) gives

(3.31) 2<> (^ + ^ ) ( ^ + ^)-^-w))

2bd(u 4- w )

in this case. The other cases can be treated similarly, see Table 2 (VII).

Case (ii). From Case (ii) of §2, we have b = 8 (mod 16) and d = 0(mod 4). Appealing to the work of Evans [5: Theorem 4 and its proof], wehave

(3.32) u = 2 (mod 4), v = 4 (mod 8), w = 2 (mod 4),

if</ΞΞθ(mod8),

and

(3.33) w = 0 (mod 4), v = 0 (mod 8), w = 0 (mod 4),

i f*/= 4 (mod 8).

If rf = 0 (mod 8), replacing g by g 1 6 ί + r (where / = 1, 7, 9, 15 and(16s + t,f) = 1), as necessary, we can suppose that

(3.34) i Ξ 8 (mod 32), w = 2 (mod 8).

There are exactly two 5-tuples (b, d, w, υ, w), which satisfy (3.13) and(3.34). These are

(b,d,u,v,w) and (b, —d9 —w, —υ9u), if« =

and

(b,d,u,υ,w) and (b, — d, w, — v, — w), ifw = 6(mod8) .

EXTENSIONS OF THEOREMS OF CUNNINGHAM-AIGNER

We note that the 16th root of unity modulo /?,

[ (dx + cv)(a(u + w)- b(u - w))

is independent of which 5-tuple is used, since

f ±u)- b( + w + u))

121

2b(-d){(+w)2 + (±u)2)jJ

where

A =

_ I (dx + cυ)(a(u + w) - b(u - w)))A

~[ 2bd(u2 + w2) J '

, if«Ξ2(mod8),

\4 ~

moreover,

- 14</ - 24M + 28H> = 0 (mod 16),

so that

= b - βd + 12M - 8w = 0 (mod 16),

= | + d - 2(M + w) (mod 16).

The choice (3.34) makes the exponent (b/4) + d — 2(u + w) in (3.26)congruent to 2 (mod 8). We now consider cases according as b =8, 40 (mod 64); d = 0, 8 (mod 16); u = 2, 6 (mod 8). For example if b =8 (mod 64), d = 0 (mod 16), u = 6 (mod 8), then (b/4) + d -2(M + w) = 2 (mod 16), so (3.26) gives

(3.35)2bd(u2 + w2)

(modp),


see Table 2(VI). We remark that in applying Theorem 2 in this case, dmust be chosen to satisfy the congruence (3.13). We can do this asx2 — 2v2 ^ 0 (mod/?), since

-p = ~x2 - 2u2 - 2υ2 - 2w2 < x2 - 2υ2 < x2 <p.

If d = 4 (mod 8), replacing g by g16*+' (where t= 1, 3, 5 or 7 and(16s + /,/) = 1), as necessary, we can suppose that

(3.36) b ΞΞ -8 ΞΞ 24 (mod 32), </ = 4 (mod 16).

There are precisely two 5-tuples (b, d, u, υ, w), which satisfy (3.13)and (3.36). These are

(b,d,u,vyw) and (b9d9 —u,v, —w).

We note that the 16th root of unity modulo p,

[ (dx + cυ)(a(u + w) - 6(n - w)) |

is independent of which 5-tuple is chosen, since

f (dx + cυ)(a(-u - w ) - b(-u + w)) |

1 2bd{(-uf+(-wf) J

CO)(O(M + w) - b(u - w))J~ 1 2bd(u2 + w2) J '

where

5 = 9 ^ + ^ + 2M + 2w) = ^ + d - 2(M + w) (mod 16).

The choice (3.36) makes the component (b/4) + d — 2(u + w) in (3.26)congruent to 2 (mod 8). We now consider cases according as b = 24, 56(mod 64); u + w = 0, 4 (mod 8). For example, if b = 56 (mod 64),M + H; = 4 (mod 8), then (6/4) + d - 2(u + w) = 10 (mod 16), so (3.26)gives

(3J7) ^ ( ^ ^ ^ y ) ) } "

(mod/?),


see Table 2(V). However, when applying Theorem 2 in this case, it is notnecessary to use the congruence bd(x2 — 2v2) = ac(u2 + 2uw — w2)(mod p) to distinguish the solutions (x, ±-u, υ9 ±w) from the solutions(x9±w9 — v9 +11). since ±w + u = ± ( w + w) (mod 8), as w = w = 0(mod 4).

Case (iii) From Case (iii) of §2 we have

(3.38) ft = 0 (mod 32), d = 4 (mod 8),

or

(3.39) ft = 16 (mod 32), d = 0 (mod 8).

If ft = 0 (mod 32), d = 4 (mod 8), from the work of Evans [5:Theorem 4 and its proof], we have

(3.40) u = 2 (mod 4), v = 4 (mod 8), w = 2 (mod 4).

Replacing g by g 1 6 ί + ' , where ί = 1, 7, 9 or 15 and (16s + / , / ) = 1, asnecessary, we can suppose that

(3.41) d = 4 (mod 16), w = 2 (mod 8).

There are exactly two 5-tuples (ft, d, u, v9 w) which satisfy (3.13) and(3.41). These are

(ft, d, w, t>, w) and ( — ft, d, — w, — υ, w), ifw = 2(mod8) ,

and

(ft, (i, w, ϋ, w) and ( — ft, d, w, —1>, — w),

We note that the 16th root of unity modulo/?,

J (dx + cυ)(a(u + w) — b(u — w))

{ 2bd(u2 + w2) " I

is independent of which 5-tuple is used, since

(dx + c( — v))(a( + w ±. u) + b( + w T U))

_ I (dx + cυ)(a(u + w) - b(u - w))) c

~{ 2bd(u2 + w2) J '


where

C = -

| , ifw = 2(mod8),

j , if w = 6 (mod 8),

and it is easily checked that

C = - + d - 2(u + w) (mod 16).

Clearly, from (3.38) and (3.40), we have (b/4) + d-2(u + w)=4(mod 8), and we determine (b/4) + d — 2(u + w) (mod 16) by consider-ing the cases b = 0, 32 (mod 64) and u = 2, 6 (mod 8). For example, ifb = 0 (mod 64) and u = 6 (mod 8), we have (b/4) + d - 2(u + w) = 4(mod 16), so by (3.26), (1.6) and (1.8),

ft 42Ϊ2bd(u2

Ξ - - (mod/?),

see Table 2 (III). In applying Theorem 2 in this case we must use thecongruence bd(x2 — 2v2) = ac(u2 -1- 2uw — w2) (mod p) to distinguishthe solutions (JC, ±w, t>, ±w) from the solutions (x, ^w, —o, ±w).

If Z? = 16 (mod 32), d = 0 (mod 8), from the work of Evans [5:Theorem 4 and its proof], we have

(3.43) w = 0(mod4), €>==0(mod8), w = 0(mod4) .

Replacing g by g l 6 5 + ί , where t—\ or 7 and (16^ + t, f) — 1, as neces-sary, we may suppose that

(3.44) b= 16 (mod 64).

There are exactly four 5-tuples (6, d, w, v9 w), which satisfy (3.13) and(3.44). These are

(fc, d, ±u, v, ±w)9 (b, -d, ±w9 -t>,

We note as before that the 16th root of unity modulo/?,

\(dx + cυ)(a(u + w)- b(u - W))γ

[ 2bd(u2 + w2) J

is independent of which 5-tuple is used.


Clearly, from (3.39) and (3.43), we have (b/4) + d~2(u + w)=4(mod 8), and we determine (b/4) + d — 2(u + w) (mod 16) by consider-ing the cases d = 0, 8 (mod 16) and u + w = 0, 4 (mod 8). For example, ifd = 0 (mod 16) and u + w = 4 (mod 8), then (b/4) + d - 2(u + w) = 12(mod 16), so by (3.26), (1.6) and (1.8),

- ί (dx + cυ)(a(u + w) -j

I 2

s + - (mod/?),

see Table 2(IV). When applying Theorem 2 in this case, we can use anyone of the four solutions (JC, ±u, v, ±w), (x, ±w9 —v, ^w), as±w ^ u =±(u + w) (mod 8).

Case (iv). As 2 ( /^ 1 )/ 1 6 = 1 (mod/?), from Table 1, we have

(3.46) b == 0 (mod 32), J = 0 (mod 8),

or

(3.47) b Ξ 16 (mod 32), d = 4 (mod 8).

If fe = 0 (mod 32), d = 0 (mod 8), appealing to the work of Evans [5:Theorem 4 and its proof], we have

(3.48) u = 0 (mod 4), v=0 (mod 8), w = 0 (mod 4).

There are exactly eight 5-tuples which satisfy (3.13) and (3.48), namely,

(b,d, ±=w,t>, ±vv), (b, -d, ±w, - D , ^W),

It is straightforward to check that

[ (dx + cp)(ιι(ιι + w) - fc(u - w))}

1 2bd(u2 + w2) J

is the same for all of these. The exponent (b/4) + d — 2(u + w) iscongruent to 0 (mod 8). It is easily determined modulo 16 by consideringthe cases b = 0, 32 (mod 64), d = 0, 8 (mod 16), and u + w = 0, 4(mod 8). For example, if b = 0 (mod 64), d = 0 (mod 16), u + w = 4(mod 8), we have Z>/4 + ί/-2(w + >v)Ξ8 (mod 16) so that, by (1.6),(1.8) and (3.26),

2 ( ,- 1 ) / 3 2 s j (dx + co)(fl(ιι 4- w) - b(u - w)) V ^ }

1 2 M ( 2 + 2 ) J V F)


see Table 2 (I). As noted by Evans [6: Comments following Theorem 7], itis unnecessary to use the congruence bd(x2 — 2v2) = ac(u2 + 2uw — w2)(mod/?) when applying Theorem 2 in this case.

Finally if b = 16 (mod 32), d = 4 (mod 8), appealing to the work ofEvans [5: Theorem 4 and its proof], we have

u = 2 (mod 4), i?Ξ=4(mod8), w = 2 (mod 4).

Replacing g by g165+/, where ί = 1, 3, 5 or 7 and (16s + t, f) = 1, asappropriate, we can choose

(3.49) b ΞΞ 16 (mod 64), J Ξ 4 (mod 16).

There are two 5-tuples (b, d, «, v9 w) satisfying (3.13) and (3.49), namely,

(b,d9 ±u, v, ±w),

and again it is easy to check that

(dx + ct))(α(w + w) — b(u — w))

2 W ( M 2 + w2)

= [ (dx + Cθ)(£l(-|/ -

1Now

j + J - 2(i/ + w) = 8 - 2(i/ + w) (mod 16)

so, by (3.26), we have

2[ - 1 , if« + w Ξ

see Table 2 (II). In applying Theorem 2 in this case, as noted by Evans [6:Comments following Theorem 7], it is necessary to use the congruencebd(x2 — 2v2) Ξ ac(u2 + 2uw — w2) (mod p). This completes the proofof Theorem 2.

4. Numerical examples, (a) p = 2113 (see Table 2 (I)). We have

(β, 6) = (33, ±32); α Ξ l ( m o d 4 ) ;

(c,</) = (-31,±24); C Ξ l ( m o d 4 ) ;

(JC,M,O,W) = ( - 1 7 , ± 2 8 , - 8 , ±8) or (-17, ± 8 , + 8 , + 2 8 ) ;

x = -\ (mod 8).


For each choice we have

b = 32 (mod 64), d = S (mod 16), u + w = 4 (mod 8),

so by Theorem 2(1), we have

Σ — Σ == — I ^ΓΠOCl ΔI 1 3 ^ .

(b) p = 257 (see Table 2 (II)). We have

(a,b) = (1, 16); a = l(mod4), ZJ Ξ 16 (mod 64);

(c, d) = (—15, 4); C Ξ I (mod4), ί/Ξ4(modl6);

(x,u,o,w) = (-9, ±6, - 4 , +6) or (-9, ± 6 , + 4 , ±6);

x = -1 (mod 8).

The congruence M(x 2 — 2v2) = ac(u2 + 2uw — w2) (mod p) is satisfiedby (x, u, v, w) — ( — 9, ±6, —4, ^6). As M + w = 0 (mod 8), by Theorem2(11), we have

2(/,-i)/32 = 28 Ξ -1 (mod 257).

(c)p = 1249 (see Table 2(111)). We have

(a,b) = (-15,32) or (-15,-32);

a = 1 (mod 4), b = 0 (mod 32)

(c, d) = (-31, -12); c = 1 (mod 4), i /^4(modl6) ;

(x,«,o,w) = (7, 10,4, -22) or (7,22,-4,10);

JC ΞΞ-l (mod8), wΞ2(mod8).

The congruence bd(x2 — 2υ2) = ac(u2 + 2uw — w2) (mod p) is satisfiedby (a, 6) = (-15, 32) and (JC, u, v, w) = (7, 22, - 4 , 10) or by(α, b) = (-15, -32) and (x, «, t), w) = (7, 10, 4, -22). Hence, byTheorem 2, taking Z> = 32, u = 22 = 6 (mod 8), we have

taking 6 = -32, u — 10 = 2 (mod 8), we have

2(/»-D/32 _ 239 = _ f e / α = 32/_ 15 = 6 6 4 ( m o d 1249).

(d) p = 1217 (see Table 2 (IV)). We have

(β,δ) = (-31,16); fl = l(mod4), b = 16 (mod 64);

(c,ί/) = (33,+8) or (33,-8); c = l(mod4);

(x,u,v,w) = (-17, ±12, - 8 , +16), (-17, ±16, +8, ±12),

x = -1 (mod 8).


As d = 8 (mod 16) and u + w = 4 (niod 8) (for each possibility), we have,by Theorem 2,

2(*-i>/32 _ 23« = -b/a = 16/31 Ξ 1139 (mod 1217).

(e) p = 577 (see Table 2 (V)). We have

(fl,6) = (l,24); a Ξ l ( m o d 4 ) , 6 = 24 (mod 32);

(c,</) = (17, -12); c Ξ l ( m o d 4 ) , </ΞΞ 4 (mod 16);

(x,«,o,w) = ( - l , ± 4 , -16, +4) or ( - 1 , ± 4 , + 1 6 , ±4).

As 6 = 24 (mod 64), M + w = 0 (mod 8), by Theorem 2(V), we have

2(,-.)/32 = 2,β = + (a + b)d^

ac 17

(f) /> = 353 (see Table 2 (VI)). We have

(α,6) = (17,8); flΞl(mod4), Z> = 8 (mod 32)

(c,</) = (-15,8) or ( - 1 5 , - 8 ) ; c = l(mod4);

(JC,II,U,W) = ( 7 , - 1 0 , - 4 , -6) or (7,-6,4,10);

x = -1 (mod 8), w Ξ 2 (mod 8).

The congruence bd(x2 — 2v2) = ac(u2 + 2uw — w2) (mod /?) is satisfiedby (c, rf) = (-15,8) and (x, u, v9 w) = (7, - 10, - 4 , -6), or by (c, rf) =(-15, -8) and (JC, w, ϋ, w) = (7, -6,4,10). Hence, by Theorem 2, takingthe first possibility, we have b = $ (mod 64), d = 8 (mod 16), w = -10 = 6(mod 8), so

= 2 π Ξ - ^ ± ^ s - ^ . s 283 (mod 353).

(g) P = 97 (see Table 2 (VIII)). We have

(α,6) = (9,-4); α Ξ l ( m o d 4 ) , 6 = 12 (mod 16);

(c, ί/) = (5, - 6 ) ; c Ξ l ( m o d 4 ) , ii = 2(mod8);

(x, u, v,w) - (7, -4,2,2); JC Ξ - 1 (mod 8), w ^ 2 ( m o d 8 ) .

As 6 = 60 (mod 64), d = 10 (mod 16), M Ξ 4 (mod 8), by Theorem 2(VII),we have

2(|>-l)/32 _ 23 = ( ~ 3 2 ) ( ~ 4 6 ) _ 23 =

(48)(20) - 1 5 -

5. Acknowledgement. We wish to thank Mr. Lee-Jeff Bell for doingsome calculations for us in connection with the preparation of this paper.


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REFERENCES

1. A. Aigner, Kriterien zum 8. und 16. Poΐenzcharakter der Reste 2 und -2, DeutscheMath., 4 (1939), 44-52.2. P. Barrucand and H. Cohn, Note on primes of type x2 + 32y 2 , class number, andresiduacity, J. Reine Angew. Math., 238 (1969), 67-70.3. A. Cunningham, On 2 as a \6-ic residue, Proc. London Math. Soc, (1) 27 (1895/6),85-122.4. G. L. Dirichlet, Ueber den biquadratischen Charakter der Zahl "Zwei", J. ReineAngew. Math., 57 (1860), 187-188.5. R. J. Evans, Resolution of sign ambiguities in Jacobi and Jacobs thai sums, Pacific J.Math., 81(1979), 71-80.6. , The 2rth power character of 2, J. Reine Angew. Math., 315 (1980), 174-189.7. R. J. Evans and J. R. Hill, The cyclotomic numbers of order sixteen, Math. Comp., 33(1979), 827-835.

8. C. F. Gauss, Theoria residuorum biquadraticorum. I, Commentationes soc. reg. sc.Gotting. recentiores., 6 (1828), 27- . II. 7 (1832), 89- , (See Untersuchungen uber HohereArithmetik, Chelsea Publ. Co. (N. Y.) (1965), 511-586.)9. H. Hasse, Der 2nte Potenzcharakter von 2 im K'όrper der 2"ten Einheitswurzeln, Rend.Circ. Mat. Palermo, Serie II, 7 (1958), 185-244.10. Emma Lehmer, On the number of solutions of uk + D = w2 (mod/?), Pacific J. Math.,5(1955), 103-118.11. , On Euler's criterion, J. Austral. Math. Soc, 1 (1959), 64-70.12. P. A. Leonard and K. S. Williams, A rational sixteenth power reciprocity law, ActaArith., 33(1977), 365-377.13. J. M. Masley and H. L. Montgomery, Cyclotomic fields with unique factorization, J.Reine Angew. Math., 286/287 (1976), 248-256.14. C. G. Reuschle, Mathematische Abhandlung, enthaltend neue Zahlentheoretische Tabel-len, Programm zum Schlusse des Schuljahrs 1855-56 am Kόniglichen Gymnasium zuStuttgart (1856), 61 pp.15. A. E. Western, Some criteria for the residues of eighth and other powers, Proc. LondonMath., (2), 9(1911), 244-272.16. A. L. Whiteman, The sixteenth power residue character of 2, Canad. J. Math., 6 (1954),364-373.17. , The cyclotomic numbers of order sixteen, Trans. Amer. Math. Soc, 86 (1957),401-413.

Received November 19, 1979 and in revised form August 18, 1981. Research supported bygrant no. A-7233 of the Natural Sciences and Engineering Research Council Canada.

UNIVERSITY OF SOUTH CAROLINA

COLUMBIA, SC U.S.A 29208

AND

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PACIFIC JOURNAL OF MATHEMATICS

EDITORS

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University of California Department of Mathematics

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University of Utah R FINN and H. SAMELSON

Salt Lake City, UT 84112 Stanford University

^ ^ * , Λ K ^ Stanford, CA 94305

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Nestor Edgardo Aguilera and Eleonor Ofelia Harboure de Aguilera, Onthe search for weighted norm inequalities for the Fourier transform . . . . . . . 1

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Bing Ren Li, The perturbation theory for linear operators of discrete type . . . . 29Peter Botta, Stephen J. Pierce and William E. Watkins, Linear

transformations that preserve the nilpotent matrices . . . . . . . . . . . . . . . . . . . . 39Frederick Ronald Cohen, Ralph Cohen, Nicholas J. Kuhn and Joseph

Alvin Neisendorfer, Bundles over configuration spaces . . . . . . . . . . . . . . . . 47Luther Bush Fuller, Trees and proto-metrizable spaces . . . . . . . . . . . . . . . . . . . . 55Giovanni P. Galdi and Salvatore Rionero, On the best conditions on the

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theorems of Cunningham-Aigner and Hasse-Evans . . . . . . . . . . . . . . . . . . . . 111John Francis Kurtzke, Jr., Centralizers of irregular elements in reductive

algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133James F. Lawrence, Lopsided sets and orthant-intersection by convex

sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Åsvald Lima, G. H. Olsen and U. Uttersrud, Intersections of M-ideals and

G-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175Wallace Smith Martindale, III and C. Robert Miers, On the iterates of

derivations of prime rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Thomas H. Pate, Jr, A characterization of a Neuberger type iteration

procedure that leads to solutions of classical boundary valueproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Carl L. Prather and Ken Shaw, Zeros of successive iterates ofmultiplier-sequence operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Billy E. Rhoades, The fine spectra for weighted mean operators . . . . . . . . . . . . 219Rudolf J. Taschner, A general version of van der Corput’s difference

theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Johannes A. Van Casteren, Operators similar to unitary or selfadjoint

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