Mathematicians hunting prime numbers discover infinite new pattern
scientificamerican.com
scientificamerican.com
https://news.virginia.edu/content/faculty-spotlight-math-pro...
The modern manifestation is mostly the intellectual product of Konrad Zuse, who wrote "digital physics" in 1969.
> https://en.wikipedia.org/wiki/Digital_physics
Wolfram, Tegmark, Bostrom, etc. are mostly downstream of Zuse.
That...? Ah, yes, that Konrad Zuse.
Fermat Reincarnation. > Fermat Reincarnation.
That’s fine for GIMPS, which only searches for Mersenne primes, but doesn’t work in general.
https://en.wikipedia.org/wiki/Primality_test#Fast_determinis... mentions several tests that do not require factorization, though.
At this point I hold one object that we agree to label "apple". Note that even seeing it as a single object is a layer of abstraction. In reality it's a clump of fundamental particles temporarily banding together
> and then handed you another apple,
What's "another apple"? What does it have in common with the thing I'm already holding? We label this thing to be also an apple, but it's a totally different set of atoms, from a different tree, perhaps from the other side of the planet. Perhaps the atoms formed in stellar processes light years away from that of the other apple.
Calling both of these things "apple" is a required first step to having two of them, but that is an of abstraction, a mental trick we use to simplify the world so we can represent it in our minds.
I'm not a particle physicist but I hear electrons *can* be counted without any unwitting help from our lower-level neural circuitry.
The numbers themselves aren't a problem, I'm just pointing out that our cognition involves many overlapping layers of abstraction, and we're doing mathematics and every other mental activity in one or more of those layers.
That this seems to correlate strongly to real-world phenomena speaks well of the types of abstraction that nature has equipped us with.
However, what about virtual things, e.g. apples in a computer game.
Addition is abstract phenomena based on math, which itself is abstract, so it can only function in abstract setup.
You are differentiating, classifying, etc.
Endless "unity...discrete" discussions have arisen in both science and philosophy since the beginning.
In any case, any distinction I have interest in or speak of would be human, whether made by a human or otherwise.
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.
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.
Pronounce "lah-BORE-i-tories", obviously.
So, n is prime iff M_1(n)=n+1. That's much simpler than the first equation listed there!
Indeed, looking things up, it seems that in general the functions M_a can be written as a linear combination (note: with polynomial coefficients, not constant) of the sigma_k (sigma_k is the sum of the k'th power of the divisors). So this result becomes a lot less surprising once you know that...
M_1(n) = sum(
m
for m in range(1, n+1)
for s in range(1, n+1)
if m*s = n
)As for the higher ones, I'm having trouble finding a proper citation saying that this was known earlier, but this math.stackexchange answer asserts that MacMahon himself worked some of this out: https://math.stackexchange.com/a/4922496/2884 No proper citation though, annoying.
When you say "this wasn't known", on what basis is that? It's very hard to be sure that something wasn't known unless you're an expert on that particular thing!
Note that the paper studies equations with polynomial coefficients on McMahon series. That is, the n+1 in our trivial observation is "stray" in a sense.
For an at-a-glance indication of nontriviality, look no further than the conjecture associated with Theorem 1.2 -- that there are exactly five equations of this sort which are prime indicators. That seems spooky, to me; I can't help but wonder what structure underlies such a small number of relations.
that's why I already got the double twin prime conjecture ready:
there exists an infinite number of consecutive twin primes. 3 examples: 11,13; 17,19. 101,103;107,109, AND 191,193;197,199... I know of another example near the 800s
there's also the dubious, or trivial, or dunno (gotta generalize this pattern as well) of the first "consecutive" twin prime but they overlap which is 3,5 and 5,7.... which reminds me of how only 2 and 3 are both primes off by one; again, I need to generalize this pattern of "last time ever primes did that"
For the triplet n, n+2, n+4, exactly one of those numbers is divisible by 3. So the only triplet n, n+2, n+4 where all numbers are prime contains 3: 3, 5, 7.
(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.
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.
Mathematicians should play with Scheme and SICP.
“In 1976, Jones, Sato, Wada, and Wiens (2) (…) produced a degree 25 polynomial in 26 variables whose positive values, as the variables vary over nonnegative integers, is the set of primes.“
https://mathoverflow.net/questions/132954/the-jones-sato-wad...
I'd have thought that was obvious.