# Hilbert number

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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 factorization)

...

693 = 9 * 77 = 21 * 33

...

1089 = 9 * 121 = 332

...

1197 = 9 * 133 = 21 * 57

...

1449 = 9 * 161 = 21 * 69

...

1617 = 21 * 77 = 33 * 49

...

1881 = 9 * 209 = 33 * 57

...

1953 = 9 * 217 = 21 * 93

...

2205 = 5 * 9 * 49 = 5 * 212

...

2277 = 9 * 253 = 33 * 69

...

2541 = 21 * 121 = 33 * 77

...

2709 = 9 * 301 = 21 * 129

...

2793 = 21 * 133 = 49 * 57

...

Numbers 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)

## References

• Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books