Blum Integer Calculator

What is a Blum Integer?

A Blum integer is a natural number formed as the product of two distinct prime numbers \( p \) and \( q \). These two primes must satisfy the following conditions:

• \( p \) and \( q \) are distinct prime numbers.
• Both \( p \) and \( q \) are of the form \( 4t + 3 \), where \( t \) is an integer. In other words, \( p \mod 4 = 3 \) and \( q \mod 4 = 3 \).

If \( n = p \times q \), and \( p \) and \( q \) meet the above criteria, then \( n \) is a Blum integer.

How to Check if a Number is a Blum Integer

1. Prime Factorization: Decompose the number into its prime factors.
2. Modulo Operation: Check if both prime factors satisfy \( p \mod 4 = 3 \) and \( q \mod 4 = 3 \).
3. If both conditions are satisfied, the number is a Blum integer. Otherwise, it is not.

Examples

Example 1: Is 15 a Blum Integer?

Example 2: Is 1909 a Blum Integer?

Example 3: Is 2023 a Blum Integer?

The First 100 Blum Integers