In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that for all n ≥ 4 {\displaystyle n\geq 4} , if p 1 , … , p n , p n + 1 {\displaystyle p_{1},\dots ,p_{n},p_{n+1}} are the first n + 1 {\displaystyle n+1} prime numbers, then
p n # = ∏ i = 1 n p i > p n + 1 2 . {\displaystyle p_{n}\#=\prod _{i=1}^{n}p_{i}>p_{n+1}^{2}.}
Barkley Rosser showed an upper bound where n # ≤ 2.83 n {\displaystyle n\#\leq 2.83^{n}} .
See also Primorial prime
Notes
References Uspensky, J. V.; Heaslet, M. A. (1939). Elementary Number Theory. New York: McGraw Hill. p. 87.