MathDB

For integers

m\geq 3

n

and

x_1,x_2, \ldots , x_m

x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n)

for every

2\leq i \leq m-1

, we say that the

m

-tuple

(x_1,x_2,\ldots , x_m)

is an arithmetic sequence in

(mod n)

. Let

p\geq 5

be a prime number and

1<a<p-1

be an integer. Let

{a_1,a_2,\ldots , a_k}

be the set of all possible remainders when positive powers of

a

are divided by

p

. Show that if a permutation of

{a_1,a_2,\ldots , a_k}

is an arithmetic sequence in

(mod p)

, then

k=p-1

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