The ideas here are the ones that, when combined together, give a general overview as how I am currently thinking of primes. The most important elements of this post remain:
– the use of digital roots to analyze potential divisibility problems,- generalization of a notion of a digital root,
– creating primes out of digital roots,
– creating (123)(332)(9442)(4421)(313) numbers based on their building blocks,
– understanding how the ordering of building blocks affects primarility.
Analogy to the proof by induction. Known induction is used for integers. Given that integers are ordered and I(n+1) = I(n) +1, we can assume that if a statement being true for I(n) implies statement being true for I(n+1), then the statement is true for all integers. This type of proof can be used for any sequence, i.e. not only the sequence of integer numbers. In particular, it can be applied to a sequence of prime numbers.
If we look for an equation true only for primes, we’d have e.g. . Lets now name , which I currently analyzed shortly for only. Could we find solutions of these polynomials? Could those help us find out more about the distribution of primes? This is the first approach.
The other one that I have in my head is that linear combinations of numbers, therein digital roots, could be used for primarility testing. Still, the generalized approach to those combinations would be required. Firstly, those don’t care about the ordering of numbers. Still, in the general form, they’d have to as, for instance, division by 4 requires last two digits, whereas division by 11 requires two conditions being met, including one with the standard digital root etc. So, a generalization of those linear combinations of nunbers could help us with the generalization of the idea of a digital root and the use of those for finding divisibility rule for any integer. That is the second approach.
The third approach is the second one but for ordered elements of the numbers. This was already handled with the notion of a generalized linear combination of numbers but, for the sake of simplification, is mentioned for the second time. The analysis of a “big” number would deploy using a specific measure for specific elements of the number (subnumbers built out of chosen digits). Such a recurrential approach could be noticed regarding the divisibility rule for 3. We have that the sum of the digits must also be divisible by 3, which is again recurrence, ie. if the original number is and the sum of its digits , then for the to be divisible by 3, must be divisible by 3, thus the sum of its digits must also be divisible by 3 etc. The same applies for $9$. Similar but with the use of two linear functions is used for the divisibility for 11. Another kindsof linear equations (that must also fit the general form) are the ones for divisibility for 19 or 17. The generic tool would have to work for all primes. Any non-prime would apply all the subsequent rules for all divisors.
And the last thing is all about the previous paragraph and some other posts I have been writing about the geometrical interpree tation of numbers.
Another thing I am going to show in the future is how the parts of numbers influence the main number, ie. for 3298472304923 how e.g. 32984 and 72304923, or 32984723049 and 23 could be used to learn about the number. In terms of iterative computation-based analysis, analytic approach as well as one other idea. This idea is as follows: when you see a number you see “ups” and “downs”, e.g. for 239482 when at 23, we are having a big “up” to 9, then a major “down” to 4, then again an up to “82” etc. Ups and downs could use different numbers of digits. The idea is to understand how to look at numbers in a simpler manner.
Also, another representation of numbers, as always. I need to understand how decimal system is hurting our coginition and liability to strive for understanding what lies beneath the numbers. But this is, again, strongly related to the analysis I had mentioned before.
Below, a couple of notes, research-related, for those who want to stay up to date with what I am doing.
Numbers with digital root = 1.
Numbers with DR=2.
Below the most latest version of my analysis. Prime numbers turned into T-Sequence: 2 3 5 7 2 4 8 1 5 2 4 1 5 7 2 8 5 7 4 8 and Xi matrix with R-sequence with light of some other frequency shed on: