Hilbert Number

The Gelfond-Schneider constant is sometimes known as the Hilbert number.

Flannery and Flannery (2000, p. 35) define a Hilbert number as a positive integer of the form  (i.e., a positive integer  such that ). The first few are then 1, 5, 9, 13, 17, 21, 25, ... (OEIS A016813). A Hilbert number  that is not divisible by a smaller Hilbert number (other than 1) is then called a Hilbert prime (or S-prime; Apostol 1976, p. 101); otherwise,  is called a Hilbert composite. The first few Hilbert primes are 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (OEIS A057948), and the first few Hilbert composites are 25, 45, 65, 81, 85, ... (OEIS A054520).

Factorization with respect to Hilbert primes is not necessarily unique, as illustrated by the example

The first few such examples are 441, 693, 1089, 1197, 1449, ... (OEIS A057949).

REFERENCES:

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, p. 101, 1976.

Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, 2000.

Sloane, N. J. A. Sequences A016813, A054520, A057948, and A057949 in "The On-Line Encyclopedia of Integer Sequences."