Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2007 / B / 5 / ii / Solution

In the P-adic numbers, ${\sum }_{j\ge 0}{p}^{2j}$ converges to $1/(1-p^2)$. ConsequentlySince $p\ne 2$, the digits one and two are both between zero and $p-1$, so no carrying is required. These are its p-adic digits, with the unique negative-power digit at exponent minus one.