Prime sequence

(3)

(3, 5)

$\left(3,5,7,\cdots ,P_n\right)$

LiKe sequence

(9), (27), $\cdots$ , (3ⁿ)

(25, 27)

$\left(Y_1,Y_2,\cdots ,Y_n\right)$

Character

3ⁿ

$25=5^2$ ;

$27=3^3$ ,

$27-25=5-3$

$Y_n=3^x{5}^y\cdots P_{n-1}^i{P}_n^j$ ;

$Y_n-Y_{n-1}=5-3$ , $\cdots$ ,

$Y_2-Y_1=P_n-P_{n-1}$

Deduction

$9>2\times \left(5-1\right)$

$25>2\times \left(7-1\right)$

$Y_1>2\times \left(P_{n+1}-1\right)$