The Aperiodicity of the Primes
Grenville Croll
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Abstract
We prove that the digits of the primes are aperiodic in all bases with a single exception. We introduce a set of related theorems that regulate the behaviour of the natural numbers through the notion of periodicity and the computational mechanism of the binary derivative. We use these theorems to establish and then investigate the behaviour of a metric p(s') which is an analytic probability of primality. This metric is based purely upon the periodicity observed in a binary number and its binary derivatives. We demonstrate that this metric is exactly quadratic. We empirically discover a small stochastic imbalance in the number of primes in the two halves of the natural numbers partitioned by their final binary derivative. We show that this stochastic imbalance must vanish in the limit such that the variance of the difference between Pi(x) and Li(x) tends to zero. This confirms our earlier work via a different method. Proof of the Riemann Hypothesis implicitly follows through the 1901 equivalence of Von Koch. We again use our metric to reorder the number line and show that the related prime density is quadratic.
Key takeaways
- We can now combine our own work with the Prime Number Theorem to give an overall estimate of the probability that a number is prime based upon its magnitude and the pattern of its binary digits.
- That is to say, as we might expect from Theorems One A&B&C and Theorem Two, most primes correspond to the values of s where none of the binary derivatives are zero i.e. zk(s) = 1 for all k. It was unexpected that there was an excess of actual over expected primes for p(s') = 0.5, where the last binary derivative is zero and the last zk(s) = 0, suggesting possible periodicity.
- Each simulation reported, for 2000 iterations, the numbers of integers that were either May Be Prime or May Not be Prime, depending upon their last binary derivative and the numbers in each of those categories which were indeed prime.
- We show that p(s') partitions the integers into quadratic sized sets where the number of primes in each set is broadly as would be expected by the Prime Number Theorem.
- We have largely constrained this analysis to the binary domain, where we have proven properties of the primes regarding their periodicities and proven properties of the binary derivative.
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Preface ix Notation xiii Frequently Used Functions xvii Chapter 1. Preliminary Notions 1 §1.1. Approximating a sum by an integral 1 §1.2. The Euler-MacLaurin formula 2 §1.3. The Abel summation formula 5 §1.4. Stieltjes integrals 7 §1.5. Slowly oscillating functions 8 §1.6. Combinatorial results 9 §1.7. The Chinese Remainder Theorem §1.8. The density of a set of integers §1.9. The Stirling formula §1.10. Basic inequalities Problems on Chapter 1 Chapter 2. Prime Numbers and Their Properties §2.1. Prime numbers and their polynomial representations §2.2. There exist infinitely many primes §2.3. A first glimpse at the size of π(x) 1 §2.4. Fermat numbers §2.5. A better lower bound for π(x) 4 iii Licensed to AMS.
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