A senary () 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 decimal, it 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 senary. Formal definition The standard set of digits in senary is given by , with a linear order . Let be the Kleene closure of , where is the operation of string concatenation for . The senary number system for natural numbers is the quotient set equipped with a shortlex order, where the equivalence class is . As has a shortlex order, it is isomorphic to the natural numbers . Mathematical properties 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 2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, ... 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: If n ≡ 0 (mod 6), 6 | n If n ≡ 2 (mod 6), 2 | n If n ≡ 3 (mod 6), 3 | n If n ≡ 4 (mod 6), 2 | 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 , where 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