Deriving the Bertrand--Chebyshev Bound from Sieve Dynamics
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| Natura: | Recurso digital |
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2025
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| _version_ | 1866901858560245760 |
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| author | Flamandzki, Artur |
| author_facet | Flamandzki, Artur |
| contents | <p>This deposition contains the preprint titled <br>“Deriving the Bertrand–Chebyshev Bound from Sieve Dynamics”. <br>The document presents a sieve-theoretic reconstruction of the classical <br>Bertrand–Chebyshev bound, explaining its origin as a structural consequence <br>of multiplicity elimination rather than analytic estimates.</p> <p>The abstract of the work is reproduced below.</p> <p>---</p> <p>The Bertrand--Chebyshev theorem states that for every natural number <br>$n>1$ the interval $(n,2n)$ contains at least one prime number. <br>Known proofs of this fact, including Erdős's proof based on the analysis <br>of binomial coefficients, confirm the existence of this bound but do not <br>describe the mechanism that makes its appearance unavoidable.</p> <p>The aim of this work is to present a sieve-based formulation in which the <br>Bertrand--Chebyshev bound arises directly from the properties of the <br>multiplicity-elimination process. The construction relies on distinguishing <br>two independent stages: <br>(i) the emergence of new prime numbers as eliminating elements, and <br>(ii) the appearance of their multiples in subsequent steps of the <br>sieving process. <br>The mutual arrangement of these two phenomena leads to identifying the point <br>at which the elimination process can no longer cover the entire considered <br>interval, thereby forcing the existence of a prime number.</p> <p>This work does not replace the classical proofs of the <br>Bertrand--Chebyshev theorem; its purpose is to explain the origin of the <br>bound itself, showing that it follows from elementary properties of how <br>the sieve operates rather than from any external assumption or additional <br>structure.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17833363 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Deriving the Bertrand--Chebyshev Bound from Sieve Dynamics Flamandzki, Artur <p>This deposition contains the preprint titled <br>“Deriving the Bertrand–Chebyshev Bound from Sieve Dynamics”. <br>The document presents a sieve-theoretic reconstruction of the classical <br>Bertrand–Chebyshev bound, explaining its origin as a structural consequence <br>of multiplicity elimination rather than analytic estimates.</p> <p>The abstract of the work is reproduced below.</p> <p>---</p> <p>The Bertrand--Chebyshev theorem states that for every natural number <br>$n>1$ the interval $(n,2n)$ contains at least one prime number. <br>Known proofs of this fact, including Erdős's proof based on the analysis <br>of binomial coefficients, confirm the existence of this bound but do not <br>describe the mechanism that makes its appearance unavoidable.</p> <p>The aim of this work is to present a sieve-based formulation in which the <br>Bertrand--Chebyshev bound arises directly from the properties of the <br>multiplicity-elimination process. The construction relies on distinguishing <br>two independent stages: <br>(i) the emergence of new prime numbers as eliminating elements, and <br>(ii) the appearance of their multiples in subsequent steps of the <br>sieving process. <br>The mutual arrangement of these two phenomena leads to identifying the point <br>at which the elimination process can no longer cover the entire considered <br>interval, thereby forcing the existence of a prime number.</p> <p>This work does not replace the classical proofs of the <br>Bertrand--Chebyshev theorem; its purpose is to explain the origin of the <br>bound itself, showing that it follows from elementary properties of how <br>the sieve operates rather than from any external assumption or additional <br>structure.</p> |
| title | Deriving the Bertrand--Chebyshev Bound from Sieve Dynamics |
| url | https://doi.org/10.5281/zenodo.17833363 |