In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... (sequence A016813 in the OEIS))
Properties The Hilbert number sequence is the arithmetic sequence with a 1 = 1 , d = 4 {\displaystyle a_{1}=1,d=4} , meaning the Hilbert numbers follow the recurrence relation a n = a n − 1 + 4 {\displaystyle a_{n}=a_{n-1}+4} . The sum of a Hilbert number amount of Hilbert numbers (1 number, 5 numbers, 9 numbers, etc.) is also a Hilbert number.
Hilbert primes A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes begins
5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (sequence A057948 in the OEIS). A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since 21 = 3 ⋅ 7. However, 21 is a Hilbert prime since neither 3 nor 7 (the only factors of 21 other than 1 and itself) are Hilbert numbers. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of the form 4n + 1 (called a Pythagorean prime), or a semiprime of the form (4a + 3) ⋅ (4b + 3).
References Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books
External links Weisstein, Eric W. "Hilbert Number". MathWorld. OEIS sequence A057949 (Numbers with more than one factorization into Hilbert primes)