I am watching fascinating videos about number theory for a while now, and I put together my own little mindworld about how prime numbers and Collatz is working. Please correct me if I’m wrong, all I have is high school maths from 15 years ago to help me.
So 1) Prime numbers.
As far as I understood, mathematicians think they are random, but my understanding of random is different, so what they mean by random is uncalculable/unpredicatable. Because if I look at the primes what they are? They are the empty spaces left by a multiplication table.
Basically 13 is a prime, because none of the previous positive integers any variation of multiplication can results those numbers.
So basically if you do a multiplication table with 2×2, 2×3, etc. with all integers, the ones missing from the results will be the primes, am I correct?
1 2 3 4 5
2 4 6 8 10
3 6 9 12 15
4 8 12 16 20
5 10 15 20 25
The numbers missing from this table are: 1, 2, 3, 5, 7, 11, 13.. basically the primes, and if do a bigger table, more primes will reveal itself. Obviously this table only capable of telling primes until the prime 5 relaibly, but this is how my thinking is.
So What is really Collatz Conjecture? It basically looking for a number which is the power of 2, (or every digit is 0 in binary except the first). If it finds one, it will go towards the bottom. If it’s not, it will go toward the botton for how many 0 digits there are (by deleting them) at the end of the binary value, then change the value by multiplying with 3 and adding 1 to get another number which will have 0 at the end. It keeps repeating this loop, until every digit is 0, except the first one, then it can reach the loop.
It means that Collatz Conjecture is a proof that every number is the sum of multiple powers of 2. for example: 11 is 1*(2^3) + 0*(2^2) + 1*(2^1) + 1*(2^0). These are basically how we write numbers in binary: 1011. Collatz is looking for a number which is the power of 2, and if it doesn’t find it, it moves to another number by multiplying 3 and adding 1. Sooner or later you will get to a power of 2 and have to the loop.
So that means that other conjectures could be built, with other base system. So I took base 3, and did the same thing. The problem is, now we have 3 variant in base 3. A number ending in 0, 1 or 2.
Obviously we are looking for a number which is ending only 0’s, except for the first digit (1000, 2000, 10000, 20000, etc), so my rule is
If the single digit sum of all digits of N are 1, 4 or 7, then N*2+1
If the single digit sum of all digits of N are 2, 5 or 8, then N*2-1
If the single digit sum of all digits of N are 3, 6 or 9, then N/3
This -I believe- will always result in the loop 3-1-3-1
My question is, do I understand these math problems right? Does anyone did another Colletz Conjecture in another base besides base 2?
Sorry for my english, it is only my second language.
Update: And how that relates to prime numbers? I had a train of thought about it, but it’s almost 4 am now.