I am unable to understand why an argument related only to p-adic number theory must be true .
Question: Assume (2.5) equivalent to equation S= -P to simplify notation( Here S and P are sums on left and right respectively in (2.5).) . Note that S+P is assumed to be rational say x/y ( x, y belongs to integers). So, as mentioned in paper I deduced that there would exist at least 1 term with -ve p-adic valuation.
But I am not convinced why on each side of series there must exists a a term with -ve p-adic valuation. In case that only one side has a p-adic valuation -ve there is a problem.
I don't have p-adic number theory in my course, but I studied article in p-adic number theory in detail from Wikipedia. Still, I am not able to apply that here and convince myself.
Could anyone please tell me the reason?
We assume $S\in \mathbb{Q}$ is not a integer, which implies there exists a prime $p$ for which $v_p(S) < 0$ (write $S$ as an irreducible fraction $S = \frac{x}{y}$, and pick any prime factor of the irreducible denominator : $p | y$). And since $P = -S$, we also get
$$v_p(P) = \underbrace{v_p(-1)}_{= 0} + v_p(S) = v_p(S) < 0$$
Now recall that $v_p\left(\sum_{i \in I} a_i\right) \ge \min_{i\in I} v_p(a_i)$ so we have
$$v_p\left(\sum_{i \in I} a_i\right)< 0 \implies \exists i\in I,\ v_p(a_i) < 0$$
So in both the sums $P$ and $S$, one of the terms must have negative valuation.