Blum number


Given a semiprime n=pq, if both p and q are Gaussian primesMathworldPlanetmath with no imaginary partMathworldPlanetmath, 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 integersMathworldPlanetmath and its prime factorsMathworldPlanetmath also have no imaginary parts. The other real semiprimes are not semiprimes among the Gaussian integers. For example, 177 can only be factored as 3×59 whether Gaussian integers are allowed or not. 159, on the other hand can be factored as either 3×53 or 3(-i)(2+7i)(7+2i).

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

Title Blum number
Canonical name BlumNumber
Date of creation 2013-03-22 17:53:17
Last modified on 2013-03-22 17:53:17
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A51
Synonym Blum integer