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Legendre's conjecture

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Legendre's conjecture is one of Landau's problems.

Conjecture. (Adrien-Marie Legendre) There is always a prime number between and provided that or 0. In terms of the prime counting function, this would mean that for all .

Jing Run Chen proved in 1975 that there is always a prime or a semiprime between and for large enough n.[1]

A natural question to ask is: Why doesn't Bertrand's postulate prove Legendre's conjecture? The reason is that actually when . For example, for , Bertrand's postulate guarantees that there is at least one prime between 9 and 18, but for Legendre's conjecture to be true we need a prime between 9 and 16. Suppose, just for the sake of argument, that 17 is prime but 11 and 13 are composite. Bertrand's postulate would still be true but Legendre's conjecture would be false. Of course the gap between and gets larger as gets larger, as A008865 shows.

Of course 11 and 13 are prime and Legendre's conjecture holds true for , and indeed it has been checked up to .

See also Brocard's conjecture.

References

  1. Jing Run Chen, On the distribution of almost primes in an interval, Sci. Sinica 18:5 (1975), pp. 611–627.