- #1
Ventrella
- 29
- 4
I identified what appears to be a partitioning of all integers > 1 into mutually disjoint sets. Each set consists of an infinite series of integers that are all the powers of what I am calling a "root" r (r is an integer that has no integer roots of its own, meaning: there is no number x^n that equals r, where x > 1 and n > 0).
For example: here are the first few integers of the first 5 sets:
2^n = 2, 4, 8, 16, 32...
3^n = 3, 9, 27, 81...
5^n = 5, 25, 125...
6^n = 6, 36...
7^n = 7, 49...
These roots include all the prime numbers, but they also include some composites. Analogous to how the primes are fundamental to multiplication, these roots are fundamental to exponentiation.
I am curious if there is an official name for these sets. Have I used proper definitions and terms? Is my assumption correct that these are mutually disjoint sets, the union of which are all the positive integers?
Thank you!
For example: here are the first few integers of the first 5 sets:
2^n = 2, 4, 8, 16, 32...
3^n = 3, 9, 27, 81...
5^n = 5, 25, 125...
6^n = 6, 36...
7^n = 7, 49...
These roots include all the prime numbers, but they also include some composites. Analogous to how the primes are fundamental to multiplication, these roots are fundamental to exponentiation.
I am curious if there is an official name for these sets. Have I used proper definitions and terms? Is my assumption correct that these are mutually disjoint sets, the union of which are all the positive integers?
Thank you!