# Is there any property of the product you can predict before using $n$-times generator $g$ $\mod P$? Can any $n$'th element have a certain property?

Given a value $$v$$ which is in same group as the generator $$g$$ modulo prime $$P$$. The group size is a prime $$s$$.
$$v = g^a \mod P$$

Only known values are $$v,g,P,s$$. Some (possible) computation of other values independent of $$v$$ is also allowed. E.g. factorization of $$P-1$$.

Now I'm looking for any (not trivial) property of $$v$$ which happen again after using $$n$$ times generator $$g$$. Or has at least a high chance. Even better if $$n$$ is the smallest number until it happen again.

Usage example:
$$g$$ is used until $$v$$ has this property. After this $$g^n$$ is used to produce the next $$v'$$ with same property.

With numbers:
For $$P$$ is not a prime there are some ways, e.g.
$$P=100, g=3, v=29$$
chosen property is 'last digit is equal to $$1$$'.
$$v=29 \not\equiv 1 \mod 10 \rightarrow 29 \cdot 3 \equiv 87 \mod 100$$
$$87 \equiv 7 \not\equiv 1 \mod 10 \rightarrow 87 \cdot 3 \equiv 61 \mod 100$$
$$61 \equiv 1 \mod 10$$ $$\checkmark$$
out of computation before we know
$$g^4 = 81 \equiv 1 \mod 10$$
with this we can predict any further value $$v''$$ has the property 'last digit is equal to $$1$$' as well
$$v'=61$$
$$v''=61 \cdot g^4 = 61 \cdot 81 \equiv 41 \mod 100$$, and $$41\equiv 1\mod 10$$ $$\checkmark$$
$$41 \cdot g^4 = 41 \cdot 81 \equiv 21 \mod 100$$, and $$21\equiv 1\mod 10$$ $$\checkmark$$
...

Now I'm looking for the case $$P$$ is a prime. Due to the fact group size $$s$$ is also a prime it's impossible for a fixed $$n$$ (except trivial $$n=s$$). So $$n$$ can depend at another (fast computable) property of $$v$$. Or only true in most cases. Another way is a min $$n_{min}$$ and a max $$n_{max}$$ to restrict the (next) appearance of that property in small region. Some (fast) calculation at $$v$$ also works.

The chosen property can be anything which can be evaluated faster than computing $$n$$ times the product with generator $$g$$ or determining the exponent of $$g$$ to produce $$v$$. The number of elements which have this property should increase with group size $$s$$ or be adaptable and $$\ll s$$.

In use case the generated group should be separated without knowing $$a$$ but same for all $$v$$ and without computing all elements.