Bonse's inequality

In number theory, Bonse's inequality, named after H. Bonse,^[1] relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if p₁, ..., p_n, p_n+1 are the smallest n + 1 prime numbers and n ≥ 4, then

${\displaystyle p_{n}\#=p_{1}\cdots p_{n}>p_{n+1}^{2}.}$

(the middle product is short-hand for the primorial ${\displaystyle p_{n}\#}$ of p_n)

Barkley Rosser showed an upper bound where ${\displaystyle n\#\leq 2.83^{n}}$.^[2]

• Primorial prime

• Uspensky, J. V.; Heaslet, M. A. (1939). Elementary Number Theory. New York: McGraw Hill. p. 87.