What does it imply if in RSA pow(m, e, n) is same as the m
Not necessarily.  With the (not so strong) primes p=211 and q=311, we have n=65621.  Using e=157, I came up with the following messages that encrypt to themselves:

1
5908
5909
11197
11817
17105
17106
23013
23014
28922
36699
42607
42608
48515
48516
53804
54424
59712
59713
65620  

Finding such a message with strong primes would be extremely difficult, but I can't think of a reason it couldn't occur.

At this point I don't have an explanation for the consecutive numbers in the list.