The congruence ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p
in a cube \mathcal B with edge length B. For a cube in general position, we show that if p \nmid abc and k \ge 5, then the congruence above has a solution in any cube with edge length B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }. Estimates are given for the case p\vert c as well, and...
Gespeichert in:
| Veröffentlicht in: | Proceedings of the American Mathematical Society 2017-02, Vol.145 (2), p.467-477 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 477 |
|---|---|
| container_issue | 2 |
| container_start_page | 467 |
| container_title | Proceedings of the American Mathematical Society |
| container_volume | 145 |
| creator | Ayyad, Anwar Cochrane, Todd |
| description | in a cube \mathcal B with edge length B. For a cube in general position, we show that if p \nmid abc and k \ge 5, then the congruence above has a solution in any cube with edge length B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }. Estimates are given for the case p\vert c as well, and improvements are given for small k. For cubes cornered at the origin, 1 \le x_i \le B for all i, we obtain a solution provided only that B\gg p^{\frac 3{2k} + O\left (\frac k{\log \log p}\right )}. Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.]]> |
| doi_str_mv | 10.1090/proc/13429 |
| format | Article |
| fullrecord | <record><control><sourceid>ams_cross</sourceid><recordid>TN_cdi_crossref_primary_10_1090_proc_13429</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1090_proc_13429</sourcerecordid><originalsourceid>FETCH-LOGICAL-a1069-a9a15c761fbd921ce492a665dae47340f61172b31fdb7a13fb69aef34404e9413</originalsourceid><addsrcrecordid>eNp9kL1OwzAYRS0EEqWw8AReWKhC_dmuk29EFX9SJRjKFslyHBtKSBNsioKqvDv9QYxMV1f36A6HkHNgV8CQjdvQ2DEIyfGADIBlWaIyrg7JgDHGE0SBx-QkxrdNBZTpgDzNXx21zfIlrNzSOmo6DZ3muS2bz0g7XdERLTq9rkbQ74L3f9uaVz3N3cdq8UUtzdu6KWl7So68eY_u7DeH5Pn2Zj69T2aPdw_T61ligClMDBqY2FSBL0rkYJ1EbpSalMbJVEjmFUDKCwG-LFIDwhcKjfNCSiYdShBDcrn_taGJMTiv27CoTfjWwPTWhd660DsXG_hiD5s6_sf9ACFNXZo</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>The congruence ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p</title><source>American Mathematical Society Publications (Freely Accessible)</source><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><source>American Mathematical Society Journals</source><source>EZB-FREE-00999 freely available EZB journals</source><creator>Ayyad, Anwar ; Cochrane, Todd</creator><creatorcontrib>Ayyad, Anwar ; Cochrane, Todd</creatorcontrib><description>in a cube \mathcal B with edge length B. For a cube in general position, we show that if p \nmid abc and k \ge 5, then the congruence above has a solution in any cube with edge length B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }. Estimates are given for the case p\vert c as well, and improvements are given for small k. For cubes cornered at the origin, 1 \le x_i \le B for all i, we obtain a solution provided only that B\gg p^{\frac 3{2k} + O\left (\frac k{\log \log p}\right )}. Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.]]></description><identifier>ISSN: 0002-9939</identifier><identifier>EISSN: 1088-6826</identifier><identifier>DOI: 10.1090/proc/13429</identifier><language>eng</language><ispartof>Proceedings of the American Mathematical Society, 2017-02, Vol.145 (2), p.467-477</ispartof><rights>Copyright 2016, American Mathematical Society</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-a1069-a9a15c761fbd921ce492a665dae47340f61172b31fdb7a13fb69aef34404e9413</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttp://www.ams.org/proc/2017-145-02/S0002-9939-2016-13429-1/S0002-9939-2016-13429-1.pdf$$EPDF$$P50$$Gams$$H</linktopdf><linktohtml>$$Uhttp://www.ams.org/proc/2017-145-02/S0002-9939-2016-13429-1/$$EHTML$$P50$$Gams$$H</linktohtml><link.rule.ids>68,69,314,780,784,23324,23328,27924,27925,77836,77838,77846,77848</link.rule.ids></links><search><creatorcontrib>Ayyad, Anwar</creatorcontrib><creatorcontrib>Cochrane, Todd</creatorcontrib><title>The congruence ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p</title><title>Proceedings of the American Mathematical Society</title><description>in a cube \mathcal B with edge length B. For a cube in general position, we show that if p \nmid abc and k \ge 5, then the congruence above has a solution in any cube with edge length B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }. Estimates are given for the case p\vert c as well, and improvements are given for small k. For cubes cornered at the origin, 1 \le x_i \le B for all i, we obtain a solution provided only that B\gg p^{\frac 3{2k} + O\left (\frac k{\log \log p}\right )}. Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.]]></description><issn>0002-9939</issn><issn>1088-6826</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp9kL1OwzAYRS0EEqWw8AReWKhC_dmuk29EFX9SJRjKFslyHBtKSBNsioKqvDv9QYxMV1f36A6HkHNgV8CQjdvQ2DEIyfGADIBlWaIyrg7JgDHGE0SBx-QkxrdNBZTpgDzNXx21zfIlrNzSOmo6DZ3muS2bz0g7XdERLTq9rkbQ74L3f9uaVz3N3cdq8UUtzdu6KWl7So68eY_u7DeH5Pn2Zj69T2aPdw_T61ligClMDBqY2FSBL0rkYJ1EbpSalMbJVEjmFUDKCwG-LFIDwhcKjfNCSiYdShBDcrn_taGJMTiv27CoTfjWwPTWhd660DsXG_hiD5s6_sf9ACFNXZo</recordid><startdate>20170201</startdate><enddate>20170201</enddate><creator>Ayyad, Anwar</creator><creator>Cochrane, Todd</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20170201</creationdate><title>The congruence ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p</title><author>Ayyad, Anwar ; Cochrane, Todd</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a1069-a9a15c761fbd921ce492a665dae47340f61172b31fdb7a13fb69aef34404e9413</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ayyad, Anwar</creatorcontrib><creatorcontrib>Cochrane, Todd</creatorcontrib><collection>CrossRef</collection><jtitle>Proceedings of the American Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ayyad, Anwar</au><au>Cochrane, Todd</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The congruence ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p</atitle><jtitle>Proceedings of the American Mathematical Society</jtitle><date>2017-02-01</date><risdate>2017</risdate><volume>145</volume><issue>2</issue><spage>467</spage><epage>477</epage><pages>467-477</pages><issn>0002-9939</issn><eissn>1088-6826</eissn><abstract>in a cube \mathcal B with edge length B. For a cube in general position, we show that if p \nmid abc and k \ge 5, then the congruence above has a solution in any cube with edge length B \gg p^{\frac 14 + \frac 1{2(\sqrt {k} +1.95)}+ \epsilon }. Estimates are given for the case p\vert c as well, and improvements are given for small k. For cubes cornered at the origin, 1 \le x_i \le B for all i, we obtain a solution provided only that B\gg p^{\frac 3{2k} + O\left (\frac k{\log \log p}\right )}. Under the assumption of GRH best possible estimates are given. Boxes with unequal edge lengths are also discussed.]]></abstract><doi>10.1090/proc/13429</doi><tpages>11</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0002-9939 |
| ispartof | Proceedings of the American Mathematical Society, 2017-02, Vol.145 (2), p.467-477 |
| issn | 0002-9939 1088-6826 |
| language | eng |
| recordid | cdi_crossref_primary_10_1090_proc_13429 |
| source | American Mathematical Society Publications (Freely Accessible); JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; American Mathematical Society Journals; EZB-FREE-00999 freely available EZB journals |
| title | The congruence ax_1x_2\cdots x_k + bx_{k+1}x_{k+2}\cdots x_{2k} \equiv c \pmod p |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-02T16%3A36%3A38IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-ams_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=The%20congruence%20ax_1x_2%5Ccdots%20x_k%20+%20bx_%7Bk+1%7Dx_%7Bk+2%7D%5Ccdots%20x_%7B2k%7D%20%5Cequiv%20c%20%5Cpmod%20p&rft.jtitle=Proceedings%20of%20the%20American%20Mathematical%20Society&rft.au=Ayyad,%20Anwar&rft.date=2017-02-01&rft.volume=145&rft.issue=2&rft.spage=467&rft.epage=477&rft.pages=467-477&rft.issn=0002-9939&rft.eissn=1088-6826&rft_id=info:doi/10.1090/proc/13429&rft_dat=%3Cams_cross%3E10_1090_proc_13429%3C/ams_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true |