① Given a "filter set". This is a set family □ (□), which depends on a parameter □. Each set □ (□) consists of a finite number of (repeatable) integers. When □→∞, the number of elements tends to infinity. ② Given a "sieve". For each given □ (□) module □, it consists of infinite sets of different prime numbers and different residue classes, where 1 ≤□ (□).
By choosing different screening sets, screening and □, subsets with different arithmetic properties can be obtained after screening, so many number theory problems can be studied by screening methods. For example, let the parameter □ be a positive integer □, □ (□) is composed of some integers greater than 1 but not greater than □, and □ is all prime numbers. Then □ =□ (integer □≥2). So □ (□ (□), □, □) consists of all integers greater than□ but not greater than□ in □ (□), and their prime factors are all greater than□. This integer is the product of not more than □- 1 prime factors. When □=2, it is Eratosthenes screening method.
For another example, let □ and □ be positive integers. Represent the proposition with {□, □}: Every sufficiently large even number is the sum of the products of two prime factors that are not greater than □ and □ respectively. The proposition {1, 1} is basically Goldbach conjecture. For this kind of proposition, the parameter □ is even □, the set □ 1 (□) = □ (□-□), 2 ≤□≤□-2}, □ are prime numbers, □ =□ (integer □≥2). If it can be proved that there exists □(□ 1(□), □, □) for an even number larger than 0, then the proposition {□- 1, □- 1} is proved. If □ 1(□) is changed to set □2(□)={□-□, prime number □