number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, Bonse's inequality, named after H. Bonse, relates the size of a

primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...

to the smallest prime that does not appear in its

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

. It states that if ''p''₁, ..., ''p''_''n'', ''p''_''n''+1 are the smallest ''n'' + 1

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s and ''n'' ≥ 4, then :

p_n\# = p_1 \cdots  p_n > p_^2.

(the middle product is short-hand for the

p_n\#

of ''p''_''n'') Mathematician Denis Hanson showed an upper bound where

n\#\leq 3^n

Notes

References

* Theorems about prime numbers Inequalities {{numtheory-stub

See also

Notes

References