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 for all ${\displaystyle n\geq 4}$, if ${\displaystyle p_{1},\dots ,p_{n},p_{n+1}}$ are the first ${\displaystyle n+1}$ prime numbers, then

${\displaystyle p_{n}\#=\prod _{i=1}^{n}p_{i}>p_{n+1}^{2}.}$

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

See also

• Primorial prime

Notes

References

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

This number theory-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e