Solving an equation with modulo

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In your particular case, your equation is a bit weird since:

sage: p == l
True

In particular, $p = 0 \mbox{ mod } l$, so there is no hope to find any nonzero $k$ such that $p^k=1 \mbox{ mod } l$.

In the more general case, the problem you are trying to solve is called the discrete logarithm. If you had some other numbers, you could have done the following:

sage: a = Mod(1,l)
sage: b = Mod(p,l)
sage: discrete_log(a,b)

As explained before, in this particular example where $p=l$ you get ValueError: No discrete log of 1 found to base 0, which seems normal.