# Blum number

Given a semiprime $n=pq$, if both $p$ and $q$ are Gaussian primes  with no imaginary part  , then $n$ is called a Blum number. The first few Blum numbers are 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, etc., listed in A016105 of Sloane’s OEIS.

A semiprime that is a Blum number is also a semiprime among the Gaussian integers  and its prime factors  also have no imaginary parts. The other real semiprimes are not semiprimes among the Gaussian integers. For example, 177 can only be factored as $3\times 59$ whether Gaussian integers are allowed or not. 159, on the other hand can be factored as either $3\times 53$ or $3(-i)(2+7i)(7+2i)$.

Large Blum numbers had applications in cryptography prior to advances in integer factorization by means of quadratic sieves  .

Title Blum number BlumNumber 2013-03-22 17:53:17 2013-03-22 17:53:17 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A51 Blum integer