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# 1. Introduction. Let Z be the set of integers, ω = (−1 + √

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(1)

2

2

1

r

1

r

α π

3

3

1

3

r

3

πα

t

3

t

t

3

t

i

(N πt−1)/3

i

t

a+bωω

3

(a+b+1)/3

a+bω1−ω

3

2(a+1)/3

λπ

3

πλ

3

2

2

3ML

2

(p−1)/3

3ML

3ML

[291]

(2)

02

02

0

0

0

0

2

2

12

13

(3u+1)(3u−3)

q

q·

(p−1)/3

3ML

d

d

dp

2

m

0

1

2

m

i

3

i

m

0

1

2

2

2

(p−1)/3

3ML

i

i

(3)

2

p

2

i

(p−1)/3

i

p

2

0

3

2

02

02

0

0

0

(p−1)/3

i

0

0

0

0

(q−1)/3

0

0

i

p

2

2

3

0

3

2

2

−3p

2

i

2

0

6

15

21

(p−(−3

p ))/3

n

0

1

n+1

n

n−1

(p−(−3

p ))/3

(p−(−3

p ))/3

n

n

(4)

−3p

5p

2

2

2

2

0

5

3

(p−1)/3

2

2

+

m

p

p

ap

απ

3

i

+

m

0

0

0

0

0

3

0

0

3

3

m

3

3

3

m

mn

3

m1m2

1

2

3

1

3

2

3

(5)

+

i

3

i

m

0

1

2

2

m

0

0

1

2

i

i

0

1

0

1

0

1

0

1

0

1

0

1

+

0

3

4 3

4

3

3

0

+

1

2

m

21

22

0

0

1

2

1

3

2

3

k1k2−3

k1+k2

0

3

1

2

1

2

1

2

1

3

2

3

1

2

1

2

3

k1k2−3

k1+k2

0

3

1

2

1

2

0

3

(6)

0

1

2

1

2

0

3

0

0

1

2

1

2

1

2

3

1

2

3

21

3

1

2

1

2

0

3

p|m/m0

1

2

1

2

3

0

m

0

i

0

i

1

2

i

1

2

1

2

1

2

i

0

3

3

0

3

3

3

3

3

3−k1k2

k1+k2

3

k1k2−3

k1+k2

3

k1k2−3

k1+k2

3

k1k2−3

k1+k2

3

1

3

2

3

i

i

i

(7)

+

m

2

2

3

2

0

3

2

3

2

k3−9k

3k2−3

3

3

2

3

3

3

1

2

+

1

2

1

2

2

i

1

i

2

j

s

t

1

r

1

r

2

2

2

i

1

3

2

3

1

3

2

3

1

3

j

s−t

1

3

1

t 3

r

i=1

i

1

3

1

s−t 3

r

i=1

1

i

3

2

s−t 3

r

i=1

2

i

3

2

3

i

2

2

α

β

0

00

+

0

00

(p−1)/3

i

(8)

i0

0

0

0

r+s

r

s−1

(p−1)/3

i

(p−1)/3

i

i

α

0

00

3

2β 3

3

i

α

0

00

3

α

0

00

3

i−2βM

i+βM

mπ00

3

mπ00

3

00

00

00

00

3

q|m00

3

q|m00

3

3

3

3

3

3

2 3

−1

α

3

α 3

(−1)s−1

α

−(−1)s+r

r

s−1

(9)

(p−1)/3

i

0

3

i+βM

α

−1 3

i0

2

2

(p−1)/3

(p−1)/3

i

i

(p−1)/3

(p−1)/3

p

3

2

3

3

2

(p−2)/3

(p+1)/3

3

3

3

3

3

3

2 3

2

3

p

p

p

p

2

(10)

3

(p2−1)/3

## = (k + 1 + 2ω)

p(p−2)3 +p−23 +p+13

(p−2)/3

(p−2)/3

(p+1)/3

2

(p−2)/3

(p+1)/3

p

2

2

p

i

(p−1)/3

i

i

(p+1)/3

i

2

2

2

2

3

2

3

2

(p−1)/3

2

(p−1)/3

2

(p−1)/3

i

3

i

2

(p−1)/3

i

(11)

2

(p−1)/3

i

(p−1)/3

i

p

3

(p−2)/3

(p−2)/3

(p+1)/3

(p−2)/3

2(p−2)3 +p+13

(p+1)/3

p−1

(p+1)/3

2

2

02

02

0

0

(p−1)/3

i

0

0

0

0

(q−1)/3

0

0

i

(p−1)/3

i

(q+1)/3

i

(p−1)/3

(12)

(p−1)/3

i

i

0

0

0

0

(q−1)/3

0

0

i

(q+1)/3

i

p

2

2

0

2

0

## (p) if and only if X

r≡(p+1)/3 (mod 3)

r

(p+1)/3

2

−(p−2)/3

0

(p−1)/3

2

(p−1)/3

2

i

r≡i (mod 3)

(p+1)/3−r

0

1

2

(p+1)/3

(p+1)/3

0

1

2

2

0

2

1

2

(13)

0

1

(p+1)/3

0

1

(p+1)/3

−(p+1)/3

2

−(p−2)/3

3

0

1

(p+1)/3

0

1

(p+1)/3

k+1+2ωp

3

0

1

(p+1)/3

−(p+1)/3

2

−(p−2)/3

0

1

(p+1)/3

0

(p+1)/3

2

−(p−2)/3

k+1+2ωp

3

0

1

(p+1)/3

0

1

(p+1)/3

−(p+1)/3

2

−(p−2)/3

0

(p+1)/3

2

−(p−2)/3

k+1+2ωp

3

2

2

3

xx33−1+1

0

(14)

1

r

1

r

2

i

3

3

2

3

3

2

3

3

2

i

3

3

2

3

3

3

3

2

3

3

2

3

3

i

xx33+1−1

0

i

x3+1

x3−1

3

r

i=1

x3+1

x3−1

i

3

i0

00

10

20

p

p

p

p

00

p

0

p

10

p

1

20

p

2

00

5

5

10

5

5

20

5

5

p

p

p

p

(3−(−3p ))/2

−3p

p

p

p

p

p−1

n

p+1

n

(15)

p

i0

−3p

i0

(p−(−3p ))/3

3r+i

3r+i

p

−3p

p

2

(p−1)/3

2

i

(p−1)/3

(p−1)/3

(p−1)/3

i(p−1)/3

(p−1)/3

(p−1)/3

3r+i

(p−1)/3

3r+i

3r+i

(p+1)/3

2

i

(p+1)/3

(p+1)/3

(p+1)/3

i(p+1)/3

(p+1)/3

(p+1)/3

3r+i

(p+1)/3

3r+i

3r+i

(p−(−3p ))/3

3·0+0

3·0+0

p

p

3r1+i

3r1+i

3r1+i

3r2+i

3r2+i

3r2+i

1

2

−3p

(16)

p

k∈C1(p)∩Rp

p

−3p

k∈C1(p)∩Rp

m

m−1

r=0

3r+1

3r+1

m

m−1

r=0

3r+1

m−1

r=0

3r+1

m−1

r=0

3r+1

m

m−1

s=0

3r+1

s

m−1

r=0

m

m−1

s=0

s

3sr

m

m−1

s=0

s

m−1

r=0

3sr

m

m−1

s=1

s

3sm

3s

m

k∈C1(p)∩Rp

m

m

m

2m

2m

m

m

2m

3m

m

3m

p

1

p

2

1

p

2

−3p

(17)

p

1

p

r

−3p

r

(p−(−3p ))/3

3r+1

3r+1

2r

(p−(−3p ))/3

(2(p−(−3p )))/3

3r+1

2

3r+1

3r+1

3r+1

3r+1

2

r2

2r

(p−1)/2

(p−1)/2

p−12 (3r+1)

3r+1

3r+1

p

p−(−3 p )

2 (3r+1)

(−3 p )−1

2 (3r+1)

3r+1

(3r+1)p

3r+1

(−3 p )−1

2 (3r+1)

3r+1

(−3p )(3r+1)

r+1

2n2

22n+1

−3p

p

0

p

00

10

20

p

0

p

0

0

p

p

p

p

p

p

00

10

20

−3p

00

−3p

00

10

20

00

p

00

10

20

r

p

r

p

(p−(−3p ))/3

r

r

p

(18)

i

j

i

j

p

(p−(−3p ))/3

2

i

i

j

j

(p−(−3p ))/3

i

i

j

j

p

(p−(−3p ))/3

i

j

i

j

i

j

i

j

p

(p−(−3p ))/3

i+j

i+j

p

i

p

j

p

i

j

i

j

p

hi+ji

p

−3p

00

10

20

1

p

00

3

p

00

10

20

00

00

3

2

p

2

3

2

p

p

00

p

p

00

p

p

p

p

p

00

10

20

p

2

p

p

3

2

p

2

i

p

i0

p

00

p

i

i

p

p

i

p

00

i0

(19)

i0

i

p

00

p

p

00

00

10

20

3

2

p

3

2

2

2

3

3

2

3

2

3

2

3

3

p

p

2

0

3

2

2

0

2

3

0

3

2

2

0

3

2

3

2

(20)

3(k−s)s

2

3

3

2

2

2

2

3

2

2

2

3

3

2

2

0

3

2

2

0

(p−(−3p ))/3

3

3

k−rk+r

3

3

3

2

3

2

3

2

3

3

3

2

2

3

3

2

2

2

2

3

3

−1

3

(21)

p

2

2

3

0

0

1

6

1

6

2

3

2

2

3

2

2

3

2

2

3

3

p

3

2

2

2

1

3

2

3

3

2

2

3

2

2

2

3

−3(b

2−4a)

p

−3(b2p−4a)

2

2

2

2

2

0

1

2

3

0

3

2

2

2

1

2

2

1

p

−3(b

2−4a)

p

3

(22)

(p−(−3p ))/3

(p+2(−3

p ))/3

n

0

1

n+1

n

n−1

p

3

p

p

−3p

p

3

p

p

3

p

p

p

3

p

3

p

2

2

2

1

2

2

p

1

−3p

p

3

2

p

p

3

2

p

2

2

−3p

2

3

2

3

2

3

(23)

3

2

p

p

3

2

3

p

p

3

2

p

p

p

3

2

p

2

2

p

3

2

2

2

2

2

p

3

p

3

p

2

2

2

1

p

2

2

1

2

p

2

1

−3p

dp

i

2

2

2

2

α

r

1

1

1

β

s

1

1

1

## ), (a, d + 3) = 1 and a(d + 3) 6≡ 0 (mod p), then

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