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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Wiley
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 |
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| _version_ | 1832545677038583808 |
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| author | George Giordano |
| author_facet | George Giordano |
| author_sort | George Giordano |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d6364bfc37fe46378759aa6b4d88ce6b |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1987-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d6364bfc37fe46378759aa6b4d88ce6b2025-02-03T07:25:03ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251987-01-0110482182310.1155/S0161171287000917The generalization and proof of Bertrand's postulateGeorge Giordano0Department of Mathematics Physics and Computer Science, Ryerson Polytechnical Institute, Toronto M5B 2K3, Ontario, CanadaThe 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.http://dx.doi.org/10.1155/S0161171287000917Bertrand's postulateprimesintervalsexplicit bound for one prime in an interval. |
| spellingShingle | George Giordano The generalization and proof of Bertrand's postulate International Journal of Mathematics and Mathematical Sciences Bertrand's postulate primes intervals explicit bound for one prime in an interval. |
| title | The generalization and proof of Bertrand's postulate |
| title_full | The generalization and proof of Bertrand's postulate |
| title_fullStr | The generalization and proof of Bertrand's postulate |
| title_full_unstemmed | The generalization and proof of Bertrand's postulate |
| title_short | The generalization and proof of Bertrand's postulate |
| title_sort | generalization and proof of bertrand s postulate |
| topic | Bertrand's postulate primes intervals explicit bound for one prime in an interval. |
| url | http://dx.doi.org/10.1155/S0161171287000917 |
| work_keys_str_mv | AT georgegiordano thegeneralizationandproofofbertrandspostulate AT georgegiordano generalizationandproofofbertrandspostulate |