81–90 of 106 posts
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Wouldn't it be fun if someone out there already knows a simple way to determine if a number is prime without factoring, but to them it is so obvious that they didn't even consider others may be interested.
Well I have a really elegant proof for this but I don't have enough space in the HN reply box to write it out -- but it is trivial, I am sure you will work it out. Fermat Reincarnation.
> Fermat Reincarnation.
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A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.
You could not be more wrong, and I am in a very bad mood unrelatedly, so that is all I will say.
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I'm not a physicist, but I think those are the smallest units in the sense that they are the smallest units we could theoretically interact with/measure, not some hard limit. It's just that it's moot to consider anything smaller because there's no way for us to ever know.
Is that because we see no way to bootstrap equipment down so many magnitudes of scale, not even close - or is it something else?
It may be that one day we come up with a more refined model. But as of today, it’s not clear how that would happen or if it’s even possible.
Imagine going from 4K to 8k to 16k resolution and then beyond. At some point a “pixel” to represent part of an image doesn’t make sense anymore, but what do you use instead? Nobody currently knows.
Because the article doesn't actually say so (presumably because the author doesn't know the difference between "if" and "if and only if") the statement: (3n^3 − 13n^2 + 18n − 8)M_1(n) + (12n^2 − 120n + 212)M_2(n) − 960M_3(n) = 0 is equivalent to the statement that n is prime. The result is that there are infinitely many such characterizing equations.
A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.
What is an arbitrary A?
This sort of thing makes me feel there is some deep understanding of reality only inches away from us, we glimpse it through these patterns but the secret remains hidden.
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Is that because we see no way to bootstrap equipment down so many magnitudes of scale, not even close - or is it something else?
Any given model has less fidelity than reality. An atlas map of the US has less detail than the actual terrain. The Planck constants represent the maximal fidelity possible with the standard model of physics. We can’t model shorter timeframes or smaller sizes, so we can’t predict what happens at scales that small. Building equipment the can measure something so small is difficult too… how do you measure something whe…
It may also be that "space" and "time" are emergent properties, much like an "apple" is "just" a description of a particular conglomeration of molecules. If we get past Planck scales it may turn that out that there are no such things as "space" and "time" and the Planck constants are irrelevant. We currently don't know but there _are_ a few theoretical frameworks that have yet to be empirically verified, like string theory.
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Can you elaborate? How does this result become less surprising if you know that? Personally I would not have guessed that there are infinitely many characterisations of P involving sums-of-powers-of-divisors either.
I mean, if you can do something a simple way, it's not that surprising that you can also do it a complicated way, I'd say.
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I was referring to spacetime in GR is modeled as smooth continuous manifold. In case you're serious though, planck length are not some fine-grained pixels/voxels in the cartesian 3d world, at least not confirmed; in-fact planck units are derived scales.
Where could you recommend I read more about how they are derived scales?
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The very notion of discreteness depends on subjective definitions of "objects". We take concepts of objects for granted because they make interacting with the world tractable, but it's really hard to define them outside of minds.
No, discrete math is exactly the same regardless of your definition of "object". It is completely independent of that. Discrete math is important to any theoretical beings that have any concept of "objects" whatsoever. It would be mostly irrelevant to entities that have no such conception, but those entities are not writing math papers.
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A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.
I have no idea what you are trying to say. What is "this" and what is the other "this"? A and B? What is an arbitrary A?
A and B are statements within a given set of axioms whose truth value is knowable within those axioms. A could be something like "Some number k that we pick is prime", and B could be something like "k is even"
When you see in math people saying "some statement is true iff some other statement is true", that "iff" stands for "if and only if", which really just means two things hold:
1. Starting with statement A, we can prove statement B.
2. Starting with statement B, we can prove statement A.
In math shorthand, we'd write this as
1. A implies B, or just A => B
2. B => A
You need to prove both directions for you to be able to say "A iff B"
Let's try it with our example statements. Does it hold? Your intuition should be saying "absolutely not", and let's see why:
1. k prime implies k is either 2, or odd. So the statement A => B only holds when k is 2. We choose k, so this could be true in a trivial case, but does not hold in the generic case
2. k even implies k is divisible by 2, so again, the statement "k even => k prime" only holds for one trivial case and not in the generic one.
Now for the original comment. I was pointing out that just because you have some proof of A iff B, does not mean there couldn't be another, completely separate statement C, for which you can prove A iff C. These relationships have equivalence, but are nonetheless not the same (outside of a categorical sense of sameness).
Some of the most compelling math of the 20th century was showing the sameness of many different fields by finding new iff relationships.