Perfect Gaussian Integer

A perfect Gaussian integer is a Gaussian integer satisfying

(1)

where is Spira's Gaussian divisor function, which is a multiplicative function. In detail, write the prime factorization

(2)

where is a unit and the Gaussian primes are chosen in the first quadrant of the complex plane, with and . Then

(3)

The unit does not enter the product (Spira 1961).

A different convention, used by Pegg (2003), chooses from each equivalence class of divisors under multiplication by a unit the unique representative with and . A Gaussian integer is perfect in this sense when the sum of its selected proper divisors equals it. Sanchez Solovieva (2026) found the example

(4)

whose prime factorization can be written

(5)

The sum of the 32 selected divisors is , so is perfect in this sense but not in Spira's.

A Gaussian integer is even when divides it. It is norm-perfect when

(6)

where . Every perfect Gaussian integer is norm-perfect, but the converse need not hold. For example, the Gaussian prime is odd and norm-perfect, but not perfect (Ward 2008). A perfect or norm-perfect Gaussian integer is called primitive when it has no nonunit perfect or norm-perfect proper divisor, respectively.

McDaniel (1974) proved a Gaussian analog of the characterization of even perfect numbers by Euclid and Euler. The primitive even perfect Gaussian integers are precisely

			(7)
			(8)

where is a rational prime number satisfying and is a Gaussian Mersenne prime. The smallest known such exponent is . In this case,

			(9)
			(10)
			(11)

It is not known whether any odd perfect Gaussian integers exist. Ward (2008) proved that every odd norm-perfect Gaussian integer is of the form

(12)

where is an odd Gaussian prime, is odd, and and are relatively prime.