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I wondered if we know the universal constants (e.g. gravitational constant, etc.) with high enough precision, and try to find a sequence of which it is part of, or a sequence that converges to it, could we use the rule to make these sequences to predict the underlying fundamental laws?

E.g. in gravity, we have $1/r^2$, where "$2$" is oddly precise, not $2.0001$, or something. This is because of geometry.

I wondered if some integers could be extracted from the universal constants?

I would assume, the more digits of a constant is known, the less “short” characterization of sequences can be found, that converges to it.

Maybe something like this:

$$F= 3.14159265359 \cdot \frac 1{r^2}$$

Try to find a sequence associated with this constant. Let's say someone comes up with:

$$4\cdot\left(\frac11 - \frac13 + \frac15 - \frac17 + \dots\right)$$

Hm. This "$4$" is odd enough. Let's try to find some physical explanation for it!

Updated law, after something is found (e.g. $4$ is the number of dimensions that are relevant in this law):

$$F= 0.7853981633975 \cdot {4\over{r^2}}$$

Maybe it would be better, if we found something like:

$$F= 0.123412341341234 \cdot \frac{1+y}{r^2}$$

Just wanted to make this idea clearer.

So basically, by using sequences instead of constants, the sequences may give hints for the underlying physical laws that govern those constants. Maybe the constant is not eliminated, but the formulated law can have more details.

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    $\begingroup$ The specific example in this post might be missing some dimensional issues, where the constant $\pi$ can be scaled to $1$ by changing the units used to measure either lengths times or masses. But I guess the point holds in Planck units, or for other examples. $\endgroup$ – jacob1729 Sep 9 at 12:37
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    $\begingroup$ What Jacob said. There's no point doing this with constants like G or c that have dimensions, since their numerical values depend on the arbitrary units chosen. But you might get lucky doing it with dimensionless constants. $\endgroup$ – PM 2Ring Sep 9 at 15:17
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    $\begingroup$ This question (v3) appears to be essentially numerology. $\endgroup$ – Qmechanic Sep 9 at 17:58
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I think that this usually works the other way around. It's not that we see from nature that gravity goes like $r^{-2}$ and then we explain it from some theory, just looking at nature we could only ever know that gravity goes like $r^\alpha$ and $\alpha = -2\pm$error. Nature will usually never tell us when something like that is precisely an integer. Rather we have a theory that gravity goes like $r^{-2}$ and then we check that all of our observations are consistent with this. In fact, our best theory of gravity (general relativity) tells us that in general gravity does not simply go like $r^{-2}$, but rather that for weak enough gravity $r^{-2}$ is a good enough approximation.

Getting back to your question, if you think that you have some idea which generates a sequence which dictates some of the constants of nature, try it out. See if it is consistent with all past, and more importantly future, observations. As a sociological fact however, if you just find a sequence that gives you some constant in nature, I don't think most physicists will be terribly interested. For the most part we are looking for a coherent theory of the phenomenon. Because if your sequence is not the consequence of some larger theory, then the question of "why this number" simply gets replaced with "why this sequence." With that caveat aside, maybe finding the right sequence is a clue which can guide you into finding the right theory, so hey, give it a shot.

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