In mathematics an automorphic number (sometimes referred to as a circular number) is a number whose square "ends" in the same digits as number itself. For example, 5² = 25, 76² = 5776, and 890625² = 793212890625, so 5, 76 and 890625 are all automorphic numbers.

The sequence of automorphic numbers begins 1, 5, 6, 25, 76, 376, 625, 9376, ... (sequence A003226 in OEIS).

Given a k-digit automorphic number Failed to parse (Missing texvc executable; please see math/README to configure.): n>1 , an automorphic number Failed to parse (Missing texvc executable; please see math/README to configure.): n'

For k greater than 1, there are at most two automorphic numbers with k digits, one ending in 5 and one ending in 6. One of them has the form:

and the other has the form:

The sum of the two numbers is 10^k + 1. The smaller of these two numbers may be less than 10^k-1; for example with k = 4 the two numbers are 9376 and 625. In this case there is only one k digit automorphic number; the smaller number could only form a k digit automorphic number if a leading 0 were added to its digits.