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Call two rational numbers $N$-indistinguishable if they have the same $p$-adic order for every prime $p$ less than $N$. Write $\sim_N$ for the relation of being $N$-indistinguishable,

Say you are given two rational numbers $a$ and $b$ and are told that at least one of them is positive. For natural numbers $n$ and $k$, does there exist a large $N=N(n,k)$ such that if each of the following pairs of numbers are $N$-indistinguishable, then both $a$ and $b$ are positive:

$$a\sim_N \binom{n}{k}, b\sim_N \binom{n}{k+1}, a+b\sim_N \binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1} $$

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Unless I've misunderstood, the answer is "trivially no", because given any $N$ you just let $M$ be the product of the gazillionth powers of all the primes less than $N$, and then you're free to change $a$ to $a-M$.

More generally, knowing facts about valuations of numbers at some finite set of primes tells you precisely nothing about any valuation at any other place, finite or infinite: this is some fancy form of the Chinese Remainder Theorem I guess.

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