Bonse's inequality

Inequality relating the primorial to square of the next prime number

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, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then

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

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

Mathematician Denis Hanson showed an upper bound where n # ≤ 3 n {\displaystyle n\#\leq 3^{n}} .[2]

See also

  • Primorial prime

Notes

  1. ^ Bonse, H. (1907). "Über eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung". Archiv der Mathematik und Physik. 3 (12): 292–295.
  2. ^ Hanson, Denis (March 1972). "On the Product of the Primes". Canadian Mathematical Bulletin. 15 (1): 33–37. doi:10.4153/cmb-1972-007-7. ISSN 0008-4395.

References

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


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