Bonse's inequality

Bonse's inequality states that for every prime pₙ with n ≥ 4, the product of all primes up to pₙ exceeds pₙ₊₁², the square of the next prime.

Primorial: The product of all primes up to a specified prime. The left-hand side of Bonse's inequality is the primorial of pₙ.
Prime number: A positive integer greater than one with no positive divisors other than one and itself. Every factor in the inequality's product, and the next term, is prime.
Primorial function: The function sending each prime to the product of all primes no greater than it. Bonse's inequality is a lower bound on this function in terms of the following prime.
Sylvester's theorem: A theorem stating that any set of k consecutive integers greater than k contains an integer with a prime factor greater than k. It is another elementary theorem that extracts prime-size information from products and consecutive integers.
Bertrand's postulate: A theorem stating that for every integer n greater than one, a prime lies between n and 2n. Prime-gap bounds such as this help control the next prime in inequalities involving primorials.
Prime sequence: The increasing sequence of all prime numbers, conventionally denoted p₁, p₂, and onward. The indices n and n+1 identify consecutive terms in this sequence.
Primorial prime: A prime number that is one greater than a primorial. The inequality compares a primorial with a square, while primorial primes arise immediately above such products.
Prime number theorem: The theorem that the number of primes no greater than x is asymptotic to x divided by log x. It supplies the asymptotic perspective behind the growth of products of small primes.
Binomial coefficient: A coefficient counting the ways to choose a fixed-size subset from a finite set. Binomial-coefficient estimates provide a standard route to bounds involving products of primes.
Prime gap: The difference between consecutive prime numbers. The next prime's distance from pₙ affects how demanding the square bound is.

Linked from 1 page

Grimm's conjectureRelated: It illustrates how prime-product estimates can constrain divisibility arguments.