Bonse's inequality

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Short description: 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, ..., pnpn+1 are the smallest n + 1 prime numbers and n ≥ 4, then

[math]\displaystyle{ p_n\# = p_1 \cdots p_n \gt p_{n+1}^2. }[/math]

(the middle product is short-hand for the primorial [math]\displaystyle{ p_n\# }[/math] of pn)

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

See also


  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. 


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