Timeline for Trigonometry / Euclidean Geometry for natural numbers?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 12, 2019 at 14:08 | comment | added | Anixx | @orgesleka thanks! This is quite what I wanted to express. | |
| Oct 12, 2019 at 14:04 | comment | added | user6671 | @Anixx: I don't think considering the number of divisors is a "norm". However you might consider the following which comes close to what you mean: Let $X_a = $ divisors of $a$. Then you can consider the Jaccard metric for example noticing that $|X_a \cap X_b| = \tau(\gcd(a,b))$ where $\tau(a) = $ number of divisors of $a$. You get $d_J(a,b) = 1-\frac{\tau(\gcd(a,b))}{\tau(a)+\tau(b)-\tau(\gcd(a,b))}$. Defining $|a| = d_J(a,1)$ then for a prime $p$ we have $|p| = 1/2$ so scaling this metric with $\times 2$ you get primes of "norm" $1$. But I am not sure if this metric embedds in $R^n$. | |
| Oct 12, 2019 at 13:00 | comment | added | Anixx | @Noah Schweber I think this is not suitable for the metric introduced in the question, but if we consider the number of divisors of a number a norm, then the prime numbers would be the set of elements with norm equal 1, that is a unit "sphere". | |
| Oct 11, 2019 at 20:39 | comment | added | Noah Schweber | In what sense would the primes correspond to the (-2)-sphere? | |
| Oct 10, 2019 at 12:46 | history | edited | Anixx | CC BY-SA 4.0 |
edited body
|
| Oct 10, 2019 at 12:23 | history | edited | Anixx | CC BY-SA 4.0 |
added 4 characters in body
|
| Oct 10, 2019 at 12:10 | comment | added | Anixx | @orgesleka fractal dimension is a measure of how much of self-copies the set contains if it is scaled up by a factor. For instance, a cube incleases 8x if we scale it twice, so its dimension is $\ln8/\ln2=3$. For a square the dimension is $\ln 4/\ln2=2$. For a line it is $\ln 2/\ln 2=1$, for a point it is $\ln 1/\ln2=0$. If you take the set of natural numbers and scale it twice, it becomes twice less dense, so we have $\ln(1/2)/\ln2=-1$ | |
| Oct 10, 2019 at 11:41 | comment | added | user6671 | thanks for your interesting answer. do you know a reference for this? (+1) | |
| Oct 10, 2019 at 11:38 | history | answered | Anixx | CC BY-SA 4.0 |