MathDB

Given a circle with its perimeter equal to

n

(

n \in N^*

), the least positive integer

P_n

which satisfies the following condition is called the “number of the partitioned circle”: there are

P_n

points (

A_1,A_2, \ldots ,A_{P_n}

) on the circle; For any integer

m

(

1\le m\le n-1

), there always exist two points

A_i,A_j

(

1\le i,j\le P_n

), such that the length of arc

A_iA_j

is equal to

m

. Furthermore, all arcs between every two adjacent points

A_i,A_{i+1}

(

1\le i\le P_n

A_{p_n+1}=A_1

) form a sequence

T_n=(a_1,a_2,,,a_{p_n})

called the “sequence of the partitioned circle”. For example when

n=13

, the number of the partitioned circle

P_{13}

=4, the sequence of the partitioned circle

T_{13}=(1,3,2,7)

(1,2,6,4)

. Determine the values of

P_{21}

and

P_{31}

, and find a possible solution of

T_{21}

and

T_{31}

respectively.

4