Lunar arithmetic

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.

 976 + 
 978 (adding digits column-wise)

    976 ×
   876 (multiplying the digits of 976 by 8)
  444  (multiplying the digits of 976 by 4)
 333   (multiplying the digits of 976 by 3)
 34876 (adding digits column-wise)

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.

Properties of the lunar operationsEdit

A few of the elementary properties of the lunar operations are listed below.[3]

  1. The lunar addition and multiplication operations satisfy the commutative and associative laws.
  2. The lunar multiplication distributes over the lunar addition.
  3. The digit 0 is the identity under lunar addition. No non-zero number has an inverse under lunar addition.
  4. The digit 9 is the identity under lunar multiplication. No number different from 9 has an inverse under lunar multiplication.

Some standard sequencesEdit

Even numbersEdit

It may be noted that, in lunar arithmetic,   and  . The even numbers are numbers of the form  . 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.


A square number is a number of the form  . So in lunar arithmetic, the first few squares are the following.


Triangular numbersEdit

A triangular number is a number of the form  . The first few triangular lunar numbers are:



In lunar arithmetic, the first few values of the factorial   are as follows:


Prime numbersEdit

In the usual arithmetic, a prime number is defined as a number   whose only possible factorisation is  . Analogously, in the lunar arithmetic, a prime number is defined as a number   whose only factorisation is   where 9 is the multiplicative identity which corresponds to 1 in usual arithmetic. Accordingly, the following are the first few prime numbers in lunar arithmetic:


Every number of the form  , where   is arbitrary, is a prime in lunar arithmetic. Since   is arbitrary this shows that there are an infinite number of primes in lunar arithmetic.

Sumsets and lunar multiplicationEdit

There is an interesting relation between the operation of forming sumsets of subsets of nonnegative integers and lunar multiplication on binary numbers. Let   and   be nonempty subsets of the set   of nonnegative integers. The sumset   is defined by


To the set   we can associate a unique binary number   as follows. Let  . For   we define


and then we define


It has been proved that

  where the " " on the right denotes the lunar multiplication on binary numbers.[4]

Magic squares of squares using lunar arithmeticEdit

A magic square of squares is a magic square formed by squares of numbers. It is not known whether there are magic square of square 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 infinity of magic squares of squares of order 3. Here is an example:[2]


See alsoEdit


  1. ^ "A087097 Lunar primes (formerly called dismal primes)". OEIS. The OEIS Foundation. Retrieved 21 October 2021.
  2. ^ a b Woll, C (2019). "There Is a 3×3 Magic Square of Squares on the Moon—A Lot of Them, Actually". The Mathematical Intelligencer. 41: 73–76. doi:10.1007/s00283-018-09866-4. Retrieved 19 October 2021.
  3. ^ a b c Applegate, David; LeBrun, Marc; Sloane, N. J. A. (2011). "Dismal Arithmetic". Journal of Integer Sequences. 14. arXiv:1107.1130. Retrieved 20 October 2021.
  4. ^ Gal Gross (2021). "Maximally Additively Reducible Subsets of the Integers". Journal of Integer Sequences. 23 (Article 20.10.5). Retrieved 21 October 2021.

External linksEdit