User:Contribution/Limit

Purpose

An harmonic limit is a set of rational numbers whose the prime numbers into its prime factorization are right-bounded.

The goal of this page is to list several kinds of rational number limiting subsets.

Minimal prime limit

A positive rational number q belongs to the pmin-limit, called the minimal prime limit, for a given prime number pmin if and only if it can be factored into primes (with positive or negative integer exponents) of size more than or equal to p.

In other words, a positive rational number q belongs to the pmin-limit if and only if all primes of its factorization into primes are left-bounded to pmin.

Maximal prime limit

Also called harmonic limit.

A positive rational number q belongs to the pmax-limit, called the maximal prime limit, for a given prime number pmax if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to pmax.

In other words, a positive rational number q belongs to the pmax-limit if and only if all primes of its factorization into primes are right-bounded to pmax.

Minimal factor limit

A positive rational number q belongs to the fmin-factor-limit, called the minimal factor limit, for a given positive integer f if and only if the sum of the exponent absolutes of its factorization into primes is more than or equal to fmin.

In other words, a positive rational number q belongs to the fmin-limit if and only if the sum of the exponent absolutes of its factorization into primes is left-bounded to fmin.

Maximal factor limit

A positive rational number q belongs to the fmax-factor-limit, called the maximal factor limit, for a given positive integer f if and only if the sum of the exponent absolutes of its factorization into primes is less than or equal to f.

In other words, a positive rational number q belongs to the fmax-limit if and only if the sum of the exponent absolutes of its factorization into primes is right-bounded to fmax.