Quote:
Originally Posted by Viliam Furik
Спасибо! But it still leaves the question "Why that period?" hanging in the air...

Just a wild guess that:
1. residue is in effect a Chinese Remainder of residues mod all factors of these (composite) Mp
2. this is an lcm() of individual periods guided by each factor.
Code:
M11 = 23 · 89
M23 = 47 · 178481
M29 = 233 · 1103 · 2089
M37 = 223 · 616318177
M41 = 13367 · 164511353
M43 = 431 · 9719 · 2099863
M47 = 2351 · 4513 · 13264529
M53 = 6361 · 69431 · 20394401
M59 = 179951 · 3203431780337
Factorizations are individual in each case (but each factor, as well known, has a multiple of (p+1) sitting in it, so this "reduced variety" reverberates through the process and helps lcm to be a multiple of (p1) for one reason or another.
Periods may (?) be different if we use the other popular seed values S0 = 10, or S0 = 2/3 (mod Mp).