I'm making my way through Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions, where questions $8$ & $9$ of Exercise $1$ of Chapter $4$ have the following, apparently, fallacious arguments:
Irrationality of $\pi$: Suppose $\pi = a/b$, and let $p \neq 2$ be a prime not dividing $a$. Then $$0 = \sin(pbπ) = \sin(pa) = \sum_{n=0}^{\infty}(−1)^n (pa)^{2n+1}/(2n + 1)! ≡ pa\ (\text{mod}\ p^2 )$$ which is absurd, hence a contradiction.
Transcendence of $e$: Suppose $e$ is algebraic, then so is $e-1$. Let $p \neq 2$ be a prime not dividing the numerators and denominators of all the coefficients of the minimal polynomials of both $e$ and $e − 1$. Then $\left\lVert e \right\rVert_p = \left\lVert e-1 \right\rVert_p = 1$. We have: $$1 = \left\lVert e-1 \right\rVert^p_p = \left\lVert (e-1)^p \right\rVert_p = \left\lVert e^p-1-\sum_{i=1}^{p-1}\binom{p}{i}(-e^i) \right\rVert_p$$ Since the binomial coefficients in the summation are all divisible by $p$, and since $\left\lVert -e \right\rVert_p=1$, it follows that $1 = \left\lVert e^p-1 \right\rVert_p = \left\lVert \sum_{n=1}^{\infty}p^n/n! \right\rVert_p$, which is impossible since each summand has $\left\lVert \cdot\right\rVert_p<1$.
Now, in the hints given to the exercises, the hint corresponding to both these problems, i.e. the problem to locate the supposed fallacy, is given as: Non-theorem 1 is being used.
The so-called Non-theorem 1 is a false theorem which points out that for $\sum_{n=1}^{\infty}a_n$ a sum of rational numbers which converges to a rational number $\left\lVert \cdot \right\rVert_p$ and also converges to a rational number in $\left\lVert \cdot \right\rVert_{\infty}$, then the rational value of the infinite sum need not be the same in both metrics.
Therefore I suspect the fallacy arises by identifying the real number $e^p$ which arises from the real limit of $\sum_{n\geq 0}p^n/n!$ to the $p-$adic number $e^p$ which arise from the $p-$adic limit of $\sum_{n\geq 0}p^n/n!$; same for $\sin(pa)$. But I don't see a concrete difference, except in abstract theory.
Does this mean we have just proven the irrationality and transcendence of the $p-$adic numbers $\pi$ and $e$, but nothing really has been said about the real numbers $\pi$ and $e$? Can someone help me understand this thoroughly?