Base-6 and Charles Sander Peirce’s Semiotics
“Beyond the considerations already adduced, the chief advantages of one base of numeration over another consist in the simplicity with which it expresses multiples, powers, and especially reciprocals of powers of the prime numbers that in human affairs naturally occur most frequently as divisors” (Charles Sanders Peirce)
“Had six taken the place in numeration that ten has actually taken division by 3 would have been performed as easily as divisions by 5 now are, that is by doubling the number and showing the decimal point one place to the right. […] so that there would have been a marked superiority of convenience in this respect in a sextal over a decimal system of arithmetic.” (Charles Sanders Peirce)
“Moreover, the multiplication table would have been only about one third as hard to learn as it is, since in place of containing 13 easy products (those of which 2 and 5 are factors) and 15 harder products (where only 3, 4, 6, 7, 8, 9 are factors), it would have contained but 7 easy products, and only 3 hard ones (namely, 4 x 4 = 24, 4 x 5 = 32, and 5 x 5 = 41)” (Charles Sanders Peirce)
In addition to this, [Peirce] remarks that in a Base-6 system, all prime numbers except for 2 and 3 will end in either 1 or 5, making it easy to calculate the remainders after division.
See: Peirce’s Philosophy of Notations and the Trade-offs in Comparing Numeral Symbol Systems
Introduction
The senary (base-6) numeral system provides a structured framework for studying prime numbers. Rooted in modular arithmetic and inspired by Charles Peirce’s semiotic principles, senary simplifies the visualization of primes and offers computational insights. This guide explores these connections, integrating advanced filtering criteria based on 6k±1 combinations.
1. Foundations of the Senary System
1.1 What is Base-6 (Senary)?
Numbers in base-6 are written using six digits: 0, 1, 2, 3, 4, 5. Each position represents a power of 6:
- The rightmost digit represents 6^0 (units).
- The next digit represents 6^1 (sixes).
- The next represents 6^2 (thirty-sixes), and so on.
Example:
The decimal number 41 is written as 105 in senary:
41 = 1 × 36 + 0 × 6 + 5 × 1.
1.2 Modular Arithmetic and Primes
Prime numbers greater than 3 follow predictable patterns in mod 6 arithmetic:
- (1 mod 6 or -5 mod 6) = 6k+1: Primes such as 7, 13, 19.
- (-1 mod 6 or 5 mod 6) = 6k−1: Primes such as 5, 11, 17.
These residues map directly to senary numbers ending in 1 and 5, making base-6 a natural framework for exploring primes.
2. Advanced Filtering: Excluding Composite Products
2.1 Composite Patterns in 6k±1
Not all numbers of the form 6k+1 or 6k−1 are prime. Many are products of numbers in these forms:
- (6a−1)(6b−1): Yields 6k+1 number (e.g., 5×11=55).
- (6a−1)(6b+1): Yields a 6k−1 number (e.g., 5×7=35).
- (6a+1)(6b+1): Yields a 6k+1 number (e.g., 7×13=91).
So, {6k-1} – {(6a−1)(6b+1)} = {set of primes in 6k-1};
and {6k+1} – ({(6a−1)(6b−1)}+{(6a+1)(6b+1)}) = {set of primes in 6k+1}.
2.2 Filtering Example in Senary
- Example 1: 55(base 10)=131(base 6) (ends in 1). Appears as candidate for prime but is 5×11, so it’s composite.
- Example 2: 35(base 10)=55(base 6) (ends in 5). Appears as candidate for prime but is 5×7, so it’s composite.
While senary endings 1 and 5 indicate candidate primes, further checks (e.g., factoring) are needed.
3. Computational Advantages of Base-6
3.1 Efficient Filtering
Senary digits simplify the exclusion of non-prime candidates:
- Numbers ending in 0: Divisible by 6.
- Numbers ending in 2 or 4: Divisible by 2.
- Numbers ending in 3: Divisible by 3.
3.2 Enhanced Sieving Algorithms
The Sieve of Eratosthenes can be optimized for senary:
- Focus on numbers ending in 1 or 5 while avoiding residues 0, 2, 3, 4.
- Exclude composite products (6a±1)(6b±1).
This reduces the computational search space significantly.
3.3 Simplified Multiplication Table
Senary arithmetic simplifies patterns. Example multiplication table (partial):
× 1 2 3 4 5
———————–
1 1 2 3 4 5
2 2 4 10 12 14
3 3 10 13 20 23
4 4 12 20 24 32
5 5 14 23 32 41
Compact representations simplify both computation and visualization.
4. Semiotic and Historical Context
4.1 Peirce’s Semiotics
Charles Peirce highlighted key principles for notation:
- Iconicity: Senary endings 1 and 5 naturally align with prime residues 6k±1.
- Simplicity: Fewer digits streamline arithmetic and prime identification.
- Analytic Depth: Senary supports detailed exploration of prime behavior.
4.2 Historical Context
Base-6 systems have historical significance:
- Babylonian base-60 influenced modern timekeeping (60 seconds/minutes).
- Indigenous counting systems often feature base-6 due to its divisibility properties.
5. Challenges and Considerations
5.1 Length of Representations
Senary numbers are longer than decimal equivalents (e.g., 1000(base 10)=4344(base 6)).
However, computational efficiencies may outweigh this drawback.
5.2 Adoption Complexity
Transitioning to senary in binary or decimal-based systems would require significant effort. Conversion overhead may offset some computational gains.
6. Applications and Speculations
6.1 Prime Distribution Analysis
Senary’s cyclic structure can help visualize:
- Patterns in prime gaps and clusters.
- Composite exclusions via modular residues.
6.2 Algorithmic Advances
Senary-based algorithms could optimize:
- Modular sieves for 6k±1 residues.
- Compact storage of primes in specialized systems.
In current environments, conversion costs might limit such advantages.
Conclusion
Base-6 provides an elegant framework for prime exploration. By integrating modular arithmetic, filtering techniques, and Peirce’s semiotic principles, senary simplifies computation and reveals deeper patterns. This approach holds theoretical and computational promise for mathematicians and theorists alike.
