Senary

A senary (/ˈsiːnəri, ˈsɛnəri/) numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to the senary system.

The standard set of digits in the senary system is ${\displaystyle {\mathcal {D}}_{6}=\lbrace 0,1,2,3,4,5\rbrace }$, with the linear order ${\displaystyle 0<1<2<3<4<5}$. Let ${\displaystyle {\mathcal {D}}_{6}^{*}}$ be the Kleene closure of ${\displaystyle {\mathcal {D}}_{6}}$, where ${\displaystyle ab}$ is the operation of string concatenation for ${\displaystyle a,b\in {\mathcal {D}}^{*}}$. The senary number system for natural numbers ${\displaystyle {\mathcal {N}}_{6}}$ is the quotient set ${\displaystyle {\mathcal {D}}_{6}^{*}/\sim }$ equipped with a shortlex order, where the equivalence class ${\displaystyle \sim }$ is ${\displaystyle \lbrace n\in {\mathcal {D}}_{6}^{*},n\sim 0n\rbrace }$. As ${\displaystyle {\mathcal {N}}_{6}}$ has a shortlex order, it is isomorphic to the natural numbers ${\displaystyle \mathbb {N} }$.

When expressed in senary, all prime numbers other than 2 and 3 have 1 or 5 as the final digit. In senary, the prime numbers are written:

That is, for every prime number p greater than 3, one has the modular arithmetic relations that either p ≡ 1 or 5 (mod 6) (that is, 6 divides either p − 1 or p − 5); the final digit is a 1 or a 5. This is proved by contradiction.

For any integer n:

Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple divisibility tests for many numbers.

Furthermore, all even perfect numbers besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form 2^{p – 1}(2^p – 1), where 2^p − 1 is prime.

Senary is also the largest number base r that has no totatives other than 1 and r − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do.