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# 1. Introduction. If we define g(k) to be the order of the set {1

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(1)

k

k

k

k

N(k)

k

k

k

N(k)

N

2

(k−1)/k

N

k

k

446k6

1

2

s

i

1/(8k3)

k1

k2

ks

(2)

M ∈Z

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−k

i

ε

k

k

k

k

k

k

(k−1)/k

N(k)

N(k)

N(k)

N

k

k

k

k

(3)

k/2

k/2

k

k

k/2

2

k

k/2

k

k

k/2

2

k−2

k

k−2

k i=1

i

k

N

a

p

p

p

N

p|i p odd

p

N

i

p|i p odd

p

N

i

a+1

k

N(k)

k

k

(k−1)/k

(4)

k

k

k

k

N(k)

k

(k−1)/k

(k−1)/k

k

k

(k−1)/k

k−1

k

k

N(k)

k

N(k)

N(k)

N(k)

k

k

k

k

k

N(k)

k

N(k)

k

k

N(k)

k

k

k

k

N(k)

k

k

k

k

k

k

N(k)

k

k

N(k)

k

N(k)

k

N(k)

k

k

k

N(k)

N(k)

k

k

N(k)

k

N(k)

k

k

k

k

k

k

N(k)

(5)

k

k

k

k

k

k

(k)N

k

k

0

0

k

0

k

0

t

t

t

0

0

N(k)

t

N(k)

0

N(k)

1

1

0

k/(k−1)

k

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k

(k−1)/k

0

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1

(k)N

t+1

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k/(k−1)

k

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k

k

0

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8k3

446k6

446k6

(6)

t

N(k)

0

t

t

t

2

t

t

2

k

t

N(k)

N(k)

t

N(k)

N(k)

N(k)

(k−1)/k

k

t

N(k)

k

k

N(k)

k

k

k

k

k

(7)

(k−1)/k

−k

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−k

k

k

k

k

k

k

N(k)

k

k

k

N

k

k

3

k

k

k

−k

k

2

−k

(8)

k

0

0

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−k

## .

k

### [11] I. M. V i n o g r a d o v, On Waring’s problem, Ann. of Math. 36 (1935), 395–405.

DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO

WATERLOO, ONTARIO CANADA N2L 3G1

E-mail: MABENNETT@JEEVES.UWATERLOO.CA

### and in revised form on 30.11.1993 (2365)

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