• Corpus ID: 147704122

Cycles and Patterns in the Sieve of Eratosthenes

@article{Grob2019CyclesAP,
  title={Cycles and Patterns in the Sieve of Eratosthenes},
  author={Georg L. Grob and Matthias Schmitt},
  journal={arXiv: General Mathematics},
  year={2019}
}
We describe recurring patterns of numbers that survive each wave of the Sieve of Eratosthenes, including symmetries, uniform subdivisions, and quantifiable, predictive cycles that characterize their distribution across the number line. We generalize these results to numbers that are relatively prime to arbitrary sets of prime numbers and derive additional insights about the distribution of integers counted by Euler's phi-function. 

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} (x) = the number of integers ≤ x that are relatively prime to all the specifically designated primes

    Following are the notations use throughout this paper

      } as the set of discarded M -prime centers for the set M = {pa, pb, pc

        Define PM as the set of integers 0 ≤ x ≤ Πpi

          Tn (x) = the number of twin n -primes centers ≤ x

            where M is a finite or infinitely countable set of primes

              Define Pn as the set of integers 0 ≤ x ≤ Πpi, i ≤ n

                T (x) = the number of twin prime centers ≤ x

                  Define DM as the set of "discarded M -prime centers," i.e., integers x such that x ε PM and that each member of M divides either (x -1) or (x + 1)

                    T{pi} (x) = the number of twin prime centers that are ≤ x and relatively prime to the single prime, pi. (This is a special case