Question

Rewriting a power series in mod p arithmetic.

Answer 1

When I say p, I mean a prime number so that we have for all x,
\[x^{p-1}\equiv 1 \mod p\]Fermat's little theorem in other words. Now we can see this as being periodicity in the exponents,
\[x^n = x^{n-k(p-1)} = x^{n \pmod {p-1}}\] (If that sorta weird notation makes sense)
All I'm saying is in mod p the only unique exponents on anything are in the set, \(\{0, 1, ..., p-2\}\). So we can rewrite a function's power series as:

\[\sum_{n=0}^\infty a_n x^n \equiv \sum_{n=0}^{p-2} \left( \sum_{k=0}^\infty a_{pn+k} \right)x^n \mod p\]

Now, it seems fine, but it also seems like maybe these new coefficients have the potential of not converging. :X Is there something we can do about that or any ideas/thoughts?

Answer 2

For example, let's try to look at \(e^x\) in this way, since its coefficients will all be:

\[a_n = (n!)^{-1}\] which seems potentially interesting. I guess maybe it won't since after you hit p, it doesn't have an inverse in mod p. ho hum...