Threefoldness

The state of an integer k being divisible by 3.

Nomenclature Describing Threefoldness.

Given the study of number bases in this work, it seems suitable that we have some simple way of describing a number k according to its remainder when k is divided by 3. These terms are analogous to parity or evenness/oddness, with a little more complexity.

In 2015, John Volan coined terms that describe threefold relationships akin to the words that pertain to parity, i.e., “odd” and “even” meaning “not divisible by 2” and “divisible by 2” respectively. The word “trine” signifies a number k divisible by 3, alternatively, k = 0 (mod 3). A “nontrine” number is one that is not divisible by 3. Thus, the number 5 is nontrine, while 9 is trine, and this is much like saying 5 is odd and 6 even. Two further terms distinguish numbers not divisible by three. We say a number that is 1 more than a multiple of 3 (i.e., k = 3m + 1 or k = 1 (mod 3)) is “overtrine”, and one that is 1 less than a multiple of 3 (i.e., k = 3m − 1 or k = 2 (mod 3)) is “undertrine”. These distinctions are unnecessary regarding parity.

The prime 3 is regular to any trine base n (i.e., 3 is a divisor of any trine number). Perfect powers of 3 are regular in trine bases, and trine numbers k are either semicoprime or regular. Since 3 is the second smallest prime and its multiples and powers fairly commonly encountered in the real world, it would seem important to use a trine base or have some means of detecting and manipulating threefoldness in a nontrine base.

Testing Threefoldness

All bases have some intuitive form of detecting threefoldness (i.e., divisibility tests for 3).

Since 3 is a divisor of all trine bases, all that is required is to examine the last digit to see if it is a multiple of 3. For nontrine bases, we cannot use regular divisibility tests.

For overtrine bases n, 3 divides ω = (n − 1), thus we use the omega divisibility test, i.e., add all the digits of the number x; if the sum is divisible by 3 so is x. Decimal is an overtrine base; we know the Rule of 3: sum the digits of a decimal integer; if the sum is divisible by 3 so is the number; 183 is divisible by 3 since 1 + 8 + 3 = 12, clearly divisible by 3.

For undertrine bases n, 3 divides α = (n + 1), we use the alpha divisibility test, i.e., add the even and the odd digits separately; subtract the even from the odd; if the difference is divisible by 3 then the number is likewise. Octal is an undertrine base. In base 8, “4213” = 37. We know octal “4213” is divisible by 3 since 4 + 1 = 5 and 2 + 3 = 5; 5 - 5 = 0. Since 0 is a multiple of 3, so must “4213” = decimal 2187.

References

Post, “Odd Number Base”, DozensOnline, 1 September 2015, retrieved 5 February 2019.