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 if p₁, ..., p_n, p_n+1 are the smallest n + 1 prime numbers and n ≥ 4, then

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

(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]

Notes

↑ Bonse, H. (1907). "Über eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung". Archiv der Mathematik und Physik. 3 (12): 292–295.
↑ 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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[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.

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See also

Notes

References