Base Prime Notation is a prime number oriented number system that leverages the mathematical property that all prime numbers greater than 3 can be expressed in the form 6k±1. This system creates a unique representation where only prime candidates of the form 6k-1 (A) or 6k+1 (B) exist as absolute values, simplifying and optimizing the process of identifying prime numbers.

Key features:

**Focus on Prime Candidates:**Only numbers of the form 6k±1 are represented in the system.**Range Optimization:**We focus on a range of -N to N for the absolute value of either |A| OR |B| rather than the positive value of 0 to N for both A AND B.**Reduced Computational Complexity:**This approach reduces the computational complexity for finding prime numbers.

## Key Concepts:

**Prime Forms**- Form A (PF = -1): 6k – 1
- Form B (PF = +1): 6k + 1

**Polarity Factor (PF)**

The polarity factor determines which form to use for identifying prime candidates:- PF = -1: For the form 6k – 1
- PF = +1: For the form 6k + 1

**Range Selection**- The system operates within a range of -N to N.
- This symmetrical range allows for efficient sieving and comprehensive coverage of prime candidates.
- The absolute value of the index in the sequence corresponds to the value of k in the 6k±1 formula.

By using this method, we can significantly reduce the number of candidates to check for primality, streamlining the process of prime number identification and potentially opening new avenues for prime number research and applications.

## Sequence Generation

Base Prime Notation generates sequences based on the chosen form and the polarity factor:

**Form A (PF = -1):**

Index | Value | Calculation |
---|---|---|

0 | -1 | 6(0) – 1 = -1 |

1 | 5 | 6(1) – 1 = 5 |

2 | -7 | 6(-1) – 1 = -7 |

3 | 11 | 6(2) – 1 = 11 |

4 | -13 | 6(-2) – 1 = -13 |

5 | 17 | 6(3) – 1 = 17 |

6 | -19 | 6(-3) – 1 = -19 |

… | … | … |

**Form B (PF = +1):**

Index | Value | Calculation |
---|---|---|

0 | 1 | 6(0) + 1 = 1 |

1 | -5 | 6(-1) + 1 = -5 |

2 | 7 | 6(1) + 1 = 7 |

3 | -11 | 6(-2) + 1 = -11 |

4 | 13 | 6(2) + 1 = 13 |

5 | -17 | 6(-3) + 1 = -17 |

6 | 19 | 6(3) + 1 = 19 |

… | … | … |

## Symmetry and Dual Marking

The system utilizes symmetry for efficiency:

**Symmetry Utilization:**When a positive multiple is marked as non-prime, its corresponding negative multiple is also marked. For example, if 25 is marked as non-prime in Form A, then -25 is also marked.**Form-Specific Sieve:**By focusing on only one form (6k – 1 or 6k + 1), the candidate pool is reduced, making the sieving process more efficient. This allows checking only every sixth number instead of every number in the original sequence.

## Conclusion

Base Prime Notation simplifies prime number identification by focusing on numbers of the form 6k±1. All prime numbers greater than 3 are found within the absolute values of either Form A (6k-1) or Form B (6k+1), meaning only one form needs to be checked to find all primes when considering the range -N , N. The system’s symmetry and form-specific sieve significantly streamline the process of identifying prime numbers, theoretically reducing computational complexity in prime-related calculations.