The generalization and proof of Bertrand's postulate
The purpose of this paper is to show that for 0<r<1 one can determine explicitly an x0 such that ∀x≥x0, ∃ at least one prime between rx and x. This is a generalization of Bertrand's Postulate. Furthermore, the same procedures are used to show that if one can find upper and lower bounds fo...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
1987-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171287000917 |
| Summary: | The purpose of this paper is to show that for 0<r<1 one can determine explicitly an x0 such that ∀x≥x0, ∃ at least one prime between rx and x. This is a generalization of Bertrand's Postulate. Furthermore, the same procedures are used to show that if one can find upper and lower bounds for θ(x) whose difference is kxρ then ∃ a prime between x and x−Kxρ, where k, K>0 are constants, 0<ρ<1 and θ(x)=∑p≤xlnp, where p runs over the primes. |
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| ISSN: | 0161-1712 1687-0425 |