Legendary guess

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The Legendre's conjecture (named after the mathematician Adrien-Marie Legendre ) is a number theory statement that says that for every natural number there is at least one prime number between and .

The question of the truth value of this conjecture is one of the Landau problems  - named after Edmund Landau , who counted it among the four conjectures about prime numbers that could not be attacked at the time at the International Congress of Mathematicians in Cambridge in 1912.

The conjecture is unproven. However, it could be shown that there is always a prime or a semi- prime number between and .

Albert Ingham proved an analogous conjecture for cubic numbers : for each sufficiently large there is between and at least one prime number.

Examples

For the prime numbers confirm the conjecture.

Related

According to Brocard's conjecture (named after Henri Brocard ), there are at least four prime numbers between and for each of them, where the nth prime number (i.e.  ...). For example, between and are the five prime numbers

This assumption is also unproven.

The Danish mathematician Ludvig Oppermann (1817–1883) assumed in 1882 ( Oppermann's conjecture ) that there is at least one prime number for between and (and also between and ). This is also unproven. It follows from the conjecture that there are at least two prime numbers between and and at least two between and (one between and and one between and ), so it is a tightening of the Legendre conjecture. It also follows that is the distance between two consecutive prime numbers . Another formulation with the prime number function is:

Web links

Individual evidence

  1. Eric W. Weisstein : Landau's Problems . In: MathWorld (English).
  2. See Jing Run Chen: On the distribution of almost primes in an interval. In: Scientia Sinica 18 (1975), pp. 611-627.
  3. See Albert E. Ingham: On the difference between consecutive primes. In: The Quarterly Journal of Mathematics, Oxford Series 8 (1937), No. 1, pp. 255-266.
  4. Eric W. Weisstein : Brocard's Conjecture . In: MathWorld (English).
  5. For example Martin Aigner, Günter M. Ziegler, Proofs from THE BOOK (German: The Book of Evidence), Springer 2018, p. 12