Are there multiple ways to complete a sequence?

Question

Given sequence F as described: $$F=\{\frac{0}{2}, \frac{2}{3}, \frac{4}{5}, \frac{6}{w}, \frac{8}{11}, \frac{10}{13}, \ldots\}$$ The value of $w$ would be $7$ because all divisors are prime numbers.
($F_i = \frac{i-1}{p_i}$ where $p_i$ is the $i$-th prime)

But I would like to know if $8$ could be a possible value for $w$ because of these reasons:

We split $F$ into subsequences $G = \{F_1, F_3, F_5, \ldots\}$ and $H = \{F_2, F_4, \ldots\}$ Then $G$ satisfies $$G_i = \frac{4i}{3^i+2}\\ G = \{\frac{0}{2}, \frac{4}{5}, \frac{8}{11}, \frac{12}{29}, \ldots\}$$ Another valid formula would be $G_i = \frac{4i}{q_{i-1}+3i}$ where $q_i$ denotes the denominator of $G_i$. This would yield $$G = \{\frac{0}{2}, \frac{4}{5}, \frac{8}{11}, \frac{12}{20}, \ldots\}$$ $H$ satisfies $$H_i = \frac{4i+2}{r_{i-1}+5}$$ where $r_i$ is the denominator of $H_i$. $$H = \{\frac{2}{3}, \frac{6}{8}, \frac{10}{13}, \frac{14}{18}\}$$ So, not minding complexity, can we accept both $7$ and $8$ as possible values for $w$?
Or is the described derivation unacceptable as proof for that and why?

Answer 1

Firstly, a proof needs to show exactly what it claims. In this case, to claim that a particular single value is the answer, it means also to claim that no other value is an answer. So an example that demonstrates one formula that gives $8$ as an answer does not at all exclude the possibility of other formulae that give different answers. Hence there is no proof. For such questions, in fact, there can be no such thing as a proof of the answer, even if everyone accepts the answer. There can however be proof of the weaker claim that there is a formula that has the given terms and gives a certain value at the desired position, which is just to show the formula and prove that it does so, but such proofs won't privilege the value given by the formula since it does not say anything about other values. My original answer in the comments and more details are as follows.

Strictly speaking, it can be any value you like by using any interpolating method. Of course when asked this kind of question, it is implied that we want to find a simple formula. "Simple" is subjective, but most who know about primes would disagree that your alternative is simpler than primes. So there is no right or wrong answer unless the criteria for the answer is precisely spelt out.

Specifically, if you want a certain value as the answer, you just need to interpolate through all the given points as well as the desired point, and there are in fact infinitely many functions that would work. For example, if you want a polynomial to go through $n$ points, there is exactly one polynomial of degree $n-1$ that does, and infinitely many polynomials of degree $n$ that does. And polynomials are just a small class of functions. The obvious "answer" to the sequence you gave uses the function that returns the $n$ prime given input $n$, which is not at all a polynomial.

In case you are wondering, it is actually possible to precisely specify "simple". For example one could say that the simplicity of a function on natural numbers is its Kolmogorov complexity, which you might find interesting to read about. But there are two issues with that. Firstly, it is overdoing it for such questions where there is one obvious "simple answer". Secondly, Kolmogorov complexity and other universal complexity measures in general are uncomputable, meaning that you cannot systematically determine an arbitrary object's complexity, not to say find a description that corresponds to the complexity. One may be able to do it for extremely small cases but not in general.

Incidentally, under any reasonable definition of complexity, the sequence you give will probably not be long enough for the sequence of primes to be the simplest answer! For example, one could argue that the answer could be $10$, by repeating $(2,3,5)$ each time increased by $8$. This is almost surely going to be simpler than the full definition of primes, so the prime sequence would lose out unless it is specially privileged (which happens to be the case in a mathematician's mind). Furthermore, I would guess that you need about 50 terms to be reasonably certain (but almost surely unable to prove) that the prime sequence is the simplest, because it needs to be simpler than just repeating the entire given sequence!