61–70 of 106 posts
Earlier quoted context omitted.
If I handed you 1 apple, and then handed you another apple, you wouldn't be surprised to find that you had 2 apples. The same trick works with oranges and pears.
Zero, one, infinity.
This have implications for public key cryptography?
I hope the twin prime conjecture will become a theorem during the remainder of my lifetime 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 firs…
I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma). 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 polynomi…
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.
I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma). 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 polynomi…
The M functions are the MacMahon’s partition functions (see the paper [1]). They were not known to relate to the sum of divisors. The M_a function counts partitions in a parts but weighing multiplicities in the partion. [1]: https://arxiv.org/abs/2405.06451
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!
I hope the twin prime conjecture will become a theorem during the remainder of my lifetime 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 firs…
This have implications for public key cryptography?
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.
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.
I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma). 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 polynomi…
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.
Earlier quoted context omitted.
I thought it was a smooth continuous manifold
To what extent are the Planck length and Planck second confirmed smallest discrete units?