Abstract
A previously derived set of sequences for generating prime numbers is modified by various algebraic and trigonometric manipulations. The modified sequences are then used to more efficiently generate prime numbers. The modified sequences are simplified from their original form and are much easier to implement into a computer program. It is shown by computation that the modified sequences more efficiently determine prime numbers because the modified sequences provide flexibility with their implementation. The added flexibility allows for the removal of redundancy in the overall process, which decreases the amount of data to process; thus, the efficiency of the prime number generator is significantly improved.
Highlights
In “Sequences for Determination of Prime Numbers by Elimination of Composites,” it was proven that there exists a function, termed the (a) function, that will generate all prime numbers starting with 5 up to some number (N), and the function generates composites periodically (Dugas and O’Connor, 2017)
It is worth noting that Equation 7 is a more efficient prime number generator because there are no summation operators: P ( n, m )
The results are in good agreement with the known density of primes data for powers of ten between one and one billion and the prime number theorem, which states that the density of prime numbers should decrease as (N) approaches infinity (Crandall and Pomerance, 2006)
Summary
In “Sequences for Determination of Prime Numbers by Elimination of Composites,” it was proven that there exists a function, termed the (a) function, that will generate all prime numbers starting with 5 up to some number (N), and the function generates composites periodically (Dugas and O’Connor, 2017). The (a) function is as shown in Equation 1: Using the half angle formulas for the sine squared and cosine squared terms, Equation 4 is obtained: a ( i ) = + ∑ l =1 − cos 2 (π l (4). Equation 4 can be simplified as shown in Equation 5 (Mandal and Asif, 2007): a(i) = 6i − a(i −1)
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