Determine amount of congruent numbers

Question

I found the claim in a paper that there are at max 8 integers mod $2^{130}-5$ congruent to one integer mod $2^{128}$.
$$u \pmod {2^{130}-5} \equiv g \pmod {2^{128}} \quad\text{ with }u \in U \quad \#U \le 8 $$ So $2^{130}-5$ is almost 4 times larger than $2^{128}$ and therefore $g , g + 2^{128}, g +2^{129}$ should be congruent integers to g. What are the other 5 integers?

Answer 1

This is not what the paper says! The paper refers to integers in the interval $[-2^{130}+6, 2^{130}-6]$, not integers modulo $2^{130}-5$. There are about twice as many of the former than the latter. You're missing all the negative values.

Answer 2

A fourth one (not always available) is $g+3\cdot 2^{128}$ I don't know where the other four come from.