Power savings for counting solutions to polynomial-factorial equations
<p>Let Pbe a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions <em>n</em> ≤ <em>N</em> to <em>n</em>! = <em>P</em>(<em>x</em>)which yields a power saving ov...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2023
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| _version_ | 1797109798374735872 |
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| author | Bui, HM Pratt, K Zaharescu, A |
| author_facet | Bui, HM Pratt, K Zaharescu, A |
| author_sort | Bui, HM |
| collection | OXFORD |
| description | <p>Let Pbe a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions <em>n</em> ≤ <em>N</em> to <em>n</em>! = <em>P</em>(<em>x</em>)which yields a power saving over the trivial bound. In particular, this applies to a centuryold problem of Brocard and Ramanujan. The previous best result was that the number of solutions is <em>o</em>(<em>N</em>). The proof uses techniques of Diophantine and Padé approximation.</p> |
| first_indexed | 2024-03-07T07:46:23Z |
| format | Journal article |
| id | oxford-uuid:0df29972-974c-4109-914e-5955d4399e68 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:46:23Z |
| publishDate | 2023 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:0df29972-974c-4109-914e-5955d4399e682023-06-12T08:11:24ZPower savings for counting solutions to polynomial-factorial equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:0df29972-974c-4109-914e-5955d4399e68EnglishSymplectic ElementsElsevier2023Bui, HMPratt, KZaharescu, A<p>Let Pbe a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions <em>n</em> ≤ <em>N</em> to <em>n</em>! = <em>P</em>(<em>x</em>)which yields a power saving over the trivial bound. In particular, this applies to a centuryold problem of Brocard and Ramanujan. The previous best result was that the number of solutions is <em>o</em>(<em>N</em>). The proof uses techniques of Diophantine and Padé approximation.</p> |
| spellingShingle | Bui, HM Pratt, K Zaharescu, A Power savings for counting solutions to polynomial-factorial equations |
| title | Power savings for counting solutions to polynomial-factorial equations |
| title_full | Power savings for counting solutions to polynomial-factorial equations |
| title_fullStr | Power savings for counting solutions to polynomial-factorial equations |
| title_full_unstemmed | Power savings for counting solutions to polynomial-factorial equations |
| title_short | Power savings for counting solutions to polynomial-factorial equations |
| title_sort | power savings for counting solutions to polynomial factorial equations |
| work_keys_str_mv | AT buihm powersavingsforcountingsolutionstopolynomialfactorialequations AT prattk powersavingsforcountingsolutionstopolynomialfactorialequations AT zaharescua powersavingsforcountingsolutionstopolynomialfactorialequations |