Abstract
A hallmark of prime numbers (primes) that both characterizes it away from other natural numbers and makes it a challenging preoccupation, is its staunch defiance to be expressed in terms of composites or as a formula listing all its sequence of elements. A classification approach, was mapped out, that fragments a prime into two: its last digit (trailer - reduced set of residue {1, 3, 7 and 9}) and the other digits (lead) whose value is incremented by either 1, 2 or 3 thus producing a modulo-3 arithmetic equation. The algorithm tracked both Polignac’s and modified Goldbach’s coefficients in order to explore such an open and computationally hard problem. Precisely 20,064,735,430 lower primes of digits 2 to 12 were parsed through validity test with the powers of 10 primes of Sloane's A006988. Adopting at most cubic terms of predictors (as the next logical step of Euler’s quadratic formula for primes) in multiple linear regression analysis, the generated outputs were analyzed to aid in building Akaike Information Criterion (AIC) best model with forward selection strategy. The main task was fragmented into atomic units of similar instances and types (an atom is a table of length 4,493,869 integer sequences where a database contains 30 relational tables with facilities for further reprocessing). A node, that supports parallel processing, stores 30 contiguous databases, and explores 4,044,482,100 successive integers. 513,649,226,700 lower natural numbers were explored by 127 hypothetical nodes yielding primes stored in 114,300 tables spread across 3,810 databases. Veriton S6630G computer system with 7.86GB usable memory and processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz were amongst the remarkable resources. Contrary to the apparent chaotic camouflage of primes as a bundle, the partitioned sample spaces reveal some remarkable patterns in terms of intervals of both sequence numbers and distances of separation from their immediate neighborhoods.
Highlights
Excepting the even number 2 and the only special odd prime 5, it was noticed that all primes terminate with either 1, 3, 7 or 9 as its last digit
With sample data running into 20 billion of observations and the need to capture the structure of a model in a fixed bell-like shape for the normal distribution [25], use of measures of dispersion is resorted to in order to dissect through all classes of the sample data as expressed in gaps of both Polignac and Goldbach’s coefficients
The analysis addresses uncertainty in a mathematical function that fits data with random errors
Summary
Excepting the even number 2 and the only special odd prime 5, it was noticed that all primes terminate with either 1, 3, 7 or 9 as its last digit. Euler’s quadratic function generates only prime numbers for n=0:39 and is a polynomial in terms of only one variable (var). Proving conjectures is a research area that is open for contribution with some of these problems having stagnated over many decades [20, 27] Is it possible to discover a polynomial which computes all possible prime numbers, exhaustively without skipping of some sequences, in error? The paper’s central research objectives are: To identify some basis for proving conjectures and persistently hard and open problems as regards primality testing. To search for such a polynomial function that formulates all possible prime numbers reliably with tolerable errors?
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