Lunar arithmetic, formerly called dismal arithmetic,[1][2] is a version of arithmetic in which the addition and multiplication operations on digits are defined as the max and min operations.
Thus, in lunar arithmetic, The lunar arithmetic operations on nonnegative multidigit numbers are performed as in usual arithmetic as illustrated in the following examples.
The world of lunar arithmetic is restricted to the set of nonnegative integers.
The concept of lunar arithmetic was proposed by David Applegate, Marc LeBrun, and Neil Sloane.
[3] In the general definition of lunar arithmetic, one considers numbers expressed in an arbitrary base
and define lunar arithmetic operations as the max and min operations on the digits corresponding to the chosen base.
[3] However, for simplicity, in the following discussion it will be assumed that the numbers are represented using 10 as the base.
A few of the elementary properties of the lunar operations are listed below.
The first few distinct even numbers under lunar arithmetic are listed below: These are the numbers whose digits are all less than or equal to 2.
So in lunar arithmetic, the first few squares are the following.
where 9 is the multiplicative identity which corresponds to 1 in usual arithmetic.
is arbitrary, is a prime in lunar arithmetic.
is arbitrary this shows that there are an infinite number of primes in lunar arithmetic.
There is an interesting relation between the operation of forming sumsets of subsets of nonnegative integers and lunar multiplication on binary numbers.
be nonempty subsets of the set
we can associate a unique binary number
It is not known whether there are any magic squares of squares of order 3 with the usual addition and multiplication of integers.
However, it has been observed that, if we consider the lunar arithmetic operations, there are an infinite amount of magic squares of squares of order 3.