In number theory, a Hilbert number is defined as a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert.
The integer sequence of Hilbert numbers is 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, ... (sequence A016813 in the OEIS). A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes is 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 121, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, ... (sequence A057948 in the OEIS). Note that Hilbert primes do not have to be prime numbers; for example, 21 is a composite Hilbert prime. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of form 4n + 1 (called a Pythagorean prime), or a semiprime of form (4a + 3) × (4b + 3).
Hilbert factorization
1 = unit
5 = prime
9 = prime
13 = prime
17 = prime
21 = prime
25 = 52
29 = prime
33 = prime
37 = prime
41 = prime
45 = 5 * 9
49 = prime
53 = prime
57 = prime
61 = prime
65 = 5 * 13
69 = prime
73 = prime
77 = prime
81 = 92
85 = 5 * 17
89 = prime
93 = prime
97 = prime
101 = prime
105 = 5 * 21
109 = prime
113 = prime
117 = 9 * 13
121 = prime
125 = 53
129 = prime
133 = prime
137 = prime
141 = prime
145 = 5 * 29
149 = prime
153 = 9 * 17
157 = prime
161 = prime
165 = 5 * 33
169 = 132
173 = prime
177 = prime
181 = prime
185 = 5 * 37
189 = 9 * 21
193 = prime
197 = prime
201 = prime
205 = 5 * 41
209 = prime
213 = prime
217 = prime
221 = 13 * 17
225 = 9 * 25
229 = prime
233 = prime
237 = prime
241 = prime
245 = 5 * 49
249 = prime
253 = prime
257 = prime
261 = 9 * 29
265 = 5 * 53
269 = prime
273 = 13 * 21
277 = prime
281 = prime
285 = 5 * 57
289 = 172
293 = prime
297 = 9 * 33
...
441 = 9 * 49 = 212 (the first number that has more than one Hilbert factorizations)
...
693 = 9 * 77 = 21 * 33
...
1089 = 9 * 121 = 332
...
1197 = 9 * 133 = 21 * 57
...
1617 = 21 * 77 = 33 * 49
...
Number with more than one Hilbert Factorizations are
441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2205, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3465, 3717, 3933, 3969, 4221, 4257, 4389, 4473, 4557, 4653, 4761, 4977, ... (sequence A057949 in the OEIS)