Bonse's inequality

The Bonsesche inequality is a statement about the growth of the primes. It states that the square of a prime number is less than the product of the smaller prime numbers. Found and published the inequality was from the Münster students Bonsecours in 1907. A wide audience was brought close to them through the popular science Mathematics Book Of Numbers and figures of the two mathematician Hans Rademacher ( 1892-1969 ) and Otto Toeplitz ( 1881-1940 ).

Formal is the Bonsesche inequality thus:

In the sequence of prime numbers, for each prime number from the fifth prime each square is smaller than the product of all previous primes; in formulas: For always

It is therefore

Tightening

How Rademacher and Toeplitz notice, it gives better results than the Bonsesche inequality; such as an inequality detected by Pafnuti Lvovitch Chebyshev, which means that each prime number is smaller than twice the respective predecessor prime. However, these better results prove only with powerful means of higher mathematics while Bonse required only elementary means for the proof of his inequality.

A higher degree of control even predicts a prime number between two square numbers. This is known as Legendre's conjecture, but has not yet been proven.

Credentials

• Robert J. Betts: Using Bonse 's Inequality to Find Upper Bounds on Prime Gaps. In: Journal of Integer Sequences 10, Article 07.3.8, 2007 PDF.

• Prime
• Set ( mathematics)
• Inequality