Lunar arithmetic, also called dismal arithmetic, is a system of operations on nonnegative integers in a fixed base 
, where 
is an integer. Digit
addition takes the maximum and digit
multiplication takes the minimum. For larger numbers,
addition combines corresponding digits, and multiplication
combines shifted partial products without carries (Applegate et al. 2011,
pp. 2 and 4-5).
Write 
and 
,
padding the base-
expansions with leading zeros. The output digits
are
 |  |  |
(1)
|
 |  |
(2)
|
where 
.
These operations form a commutative semiring
with additive identity element 0 and multiplicative
identity 
(Applegate et al. 2011, pp. 5-7).
A lunar prime in base 
is a number other than 
whose only factorizations
under 
have a factor equal to 
. In base 10, the first lunar primes
are 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, ... (OEIS A087097;
Applegate et al. 2011, p. 7). Every lunar prime must contain the digit 
, but this condition is not sufficient:
(Applegate et
al. 2011, p. 7).
