Bonse's inequality explained

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

p _{n\#} = p _{1} … p _{n} > p ^{2 .} _{n +1} (the middle product is short-hand for the primorial p _{n\#} of p_n)

Mathematician Denis Hanson showed an upper bound where

n\#\leq 3 ^{n} .^[2]

See also

• Primorial prime

References

• Book: Uspensky, J. V. . M. A. . Heaslet . Elementary Number Theory . McGraw Hill . New York . 1939 . 87 .

Notes and References

1. H. . Bonse . Über eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung . Archiv der Mathematik und Physik . 3 . 12 . 1907 . 292 - 295 .
2. Hanson . Denis . March 1972 . On the Product of the Primes . . 15 . 1 . 33–37 . 10.4153/cmb-1972-007-7. free . 0008-4395.