Note: By entering the caption math.SE listed a bunch of posts but none answerd my question. Also Wikipedia was no help for my case, so I dare to ask here.
The LCG $s_{k+1} = a\cdot s_k \pmod m$ in question is $$s_{k+1} = 2851130928467\cdot s_k \pmod {10^{15}}$$ Its period length is $5\cdot 10^{13}$, I was kindly told. This period length (and my limited resources) thwart to prepare a complete list of a sequence. It would take me years before I could test another seed. That is why I have three questions:
a) A period length of $5\cdot 10^{13}$ and $m=10^{15}$ determine gaps in the sequence of results. Since all results are odd only $1$ of $10$ will show up. Are there, depending on the seed, more than one sequences possible? If yes, how many do exist? If less than $10$, are "outsider" seeds attracted by a single or few more iterations to the "principal" sequence/s?
b) How may the $n^{th}$ result be predicted (computed)? I know already the $5\cdot 10^{13}$-th result, it's the seed $s_0$, but how about others? If I look at the $10^8$-th and the $2\cdot 10^8$-th results, at last the $9$ least significant digits are the same as those of the seed (the pattern known when using this kind of LCGs). Thus I surmise there could be a way to calculate any result with no need to compute all the n-1 preceding it.
c) By chance I know a seed $s_0=854559739889001$ which results in $|\Delta{s}|=378$, which is surprising low me think. I found another seed with the same $\Delta{s}$. Is there a chance to compute seeds with even a lower difference of the two following results?
