Fourfoldness

The state of an integer k being divisible by 4.

Nomenclature Describing Fourfoldness.

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 4. These terms are analogous to parity or evenness/oddness, with a little more complexity.

In 2015, John Volan coined terms that describe threefold and fourfold 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 “quadrate” signifies a number k divisible by 4, alternatively, k = 0 (mod 4). A “nonquadrate” number is one that is not divisible by 4. Thus, the number 5 is nonquadrate, while 8 is quadrate, and this is much like saying 5 is odd and 8 even. Three further terms distinguish numbers not divisible by 4. We say a number that is 1 more than a multiple of 4 (i.e., k = 4m + 1 or k = 1 (mod 4)) is “overquadrate”, and one that is 1 less than a multiple of 4 (i.e., k = 4m − 1 or k = 3 (mod 4)) is “underquadrate”. Finally, an even number not divisible by 4 is called “counterquadrate”. These distinctions are unnecessary regarding parity.

The prime 4 is regular to any quadrate or counterquadrate base n (i.e., 4 is a divisor of any quadrate number n, but divides n² for counterquadrate n). Perfect powers of 4 are regular in quadrate and counterquadrate bases, and quadrate numbers k are either semicoprime or regular. Since 4 is a small number and the smallest composite, its multiples and powers fairly commonly encountered in the real world, it might seem important to use a quadrate base or have some means of detecting and manipulating fourfoldness in a nonquadrate base.

Testing Fourfoldness

All bases have some intuitive form of detecting fourfoldness (i.e., divisibility tests for 4).

Since 4 is a divisor of all quadrate bases, all that is required is to examine the last digit to see if it is a multiple of 4. For nonquadrate bases, we cannot use regular divisibility tests.

In counterquadrate bases n, we examine the rightmost pair of digits to see if it is one of n²/4 two-digit combinations. Ten is counterquadrate, since it is even but indivisible by 4. Therefore in decimal we examine the last two digits of x to see if it is one of 25 two-digit combinations (00, 04, 08, 12, 16, …, 96). Thus, we know that 28448 is quadrate, since it has the two last digits 48. If the reference range is too long, we can examine the penultimate digit to see if it is even. If so, and the last digit is quadrate, i.e., one of (0, 4, 8, digit-12, digit-16, etc.) then it is quadrate. If the penultimate digit is odd and the last digit is counterquadrate, i.e., one of (2, 6, digit-10, digit-14, etc.), then it is quadrate.

For overquadrate bases n, 4 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 4 so is x. Nonary (base 9) is an overquadrate base. We sum the digits of a nonary integer; if the sum is divisible by 4 so is the number; “484” is divisible by 4 since 4 + 8 + 4 = “13”, clearly divisible by 4 (since 1 + 3 = 4).

For underquadrate bases n, 3 divides α = (n + 1), we use the alpha divisibility test, i.e., add the digits in even places and those in odd places separately; subtract the first from the last sum; if the difference is divisible by 4 then the number is likewise. Undecimal (base 11) is an underquadrate base. In base 11, “213” = 4⁴. We know undecimal “213” is divisible by 4 since (2 + 3) − 1 = 4.

References

Post, “Why Are Numbers Classified As Odd Or Even?”, DozensOnline, 23 November 2016, retrieved July 2019.