{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21236"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21236","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Combinatorial approaches to integer sequences","abstract":"Combinatorial methods are used to prove several results in number theory. The chapters may be read independently, and are briefly discussed below.","abstract_html":"Combinatorial methods are used to prove several results in number theory. The chapters may be read independently, and are briefly discussed below.","abstract_has_math":false,"creators":["Malouf, Janice L."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Halberstam, Heini"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:02:35Z","date_published":"2011-05-07T13:02:35Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Malouf, Janice L."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512476","(UMI)AAI9512476"],"render_values":[{"text":"AAI9512476","href":null,"code":true},{"text":"(UMI)AAI9512476","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21236","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Halberstam, Heini"]},{"key":"dc:creator","label":"Author","values":["Malouf, Janice L."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:02:35Z","10000-01-01","1994"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Malouf, Janice L."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512476","(UMI)AAI9512476","http://hdl.handle.net/2142/21236"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Combinatorial methods are used to prove several results in number theory. The chapters may be read independently, and are briefly discussed below.","In 1935 Erdos proved that every additive basis $\\rm{\\cal B}$ of order h is an essential component by establishing the inequality $\\rm \\sigma({\\cal A} + {\\cal B})\\ge \\sigma({\\cal A}) + {1\\over {2h}}\\sigma({\\cal A})(1-\\sigma({\\cal A})),$ where $\\rm\\sigma({\\cal A})$ denotes the Schnirelmann density of $\\rm{\\cal A}$. This lower bound was improved by Helmut Plunnecke in 1970 to $\\sigma({\\cal A} + {\\cal B})\\ge\\sigma ({\\cal A})\\sp{1-1/h}$ using an application of graph theory. A simplification of Plunnecke's proof is presented in Chapter 1.","The sequence of numbers $\\{ a\\sb{i}\\}$ defined by the recurrence $a\\sb{n} = (a\\sb{n-3}a\\sb{n-1} + a\\sbsp{n-2}{2})/a\\sb{n-4}$ for n $>$ 3, with initial values $a\\sb0, a\\sb1, a\\sb2, a\\sb3$ = 1, is shown to be integral in Chapter 2. The proof is extended to address more general sequences of this type.","In a famous work so entitled, Erdos and Selfridge established that the product of consecutive integers is never a power. In Chapter 3 related problems are considered in which one starts with n, not a kth power and selects a set of integers larger than n whose product with n forms a kth power, seeking to minimize the largest number used. In the restricted problem, the condition is placed on gaps between integers chosen so that no k consecutive numbers are omitted. In the case of squares, it is shown that the largest number used will not exceed 3n $-$ 3.","A set of integers is called sum-free if it contains no solution to the equation x + y = z. Erdos showed that every set of n integers has a sum-free subset with at least n/3 elements. This was strengthened by Alon and Kleitman to $>$n/3, and they showed by means of an example that the 1/3 cannot be improved to any number as large as 12/29(=.4137$\\...$). A construction is given in Chapter 4 which shows that it cannot be improved to 2/5.","A well-known theorem of Pillai and Szekeres states that for k $\\le$ 16, every set of k consecutive integers contains one which is relatively prime to the others. It was established by Brauer and Pillai that this is false for k $>$ 17. The condition of coprimality is strengthened to require that each of the numbers $\\{n,n + 1,\\..., n + k\\}$ have a factor in common with either n or n + k, and values of k for which this is possible are studied in Chapter 5.","Made available in DSpace on 2011-05-07T13:02:35Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512476.pdf: 1859542 bytes, checksum: cb9db6fc324ca7d79e98dc3352e8c2c9 (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:49:24Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:22:28-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Combinatorial approaches to integer sequences"]}]}],"canonical_facts":{"dc:contributor":["Halberstam, Heini"],"dc:creator":["Malouf, Janice L."],"dc:date":["2011-05-07T13:02:35Z","10000-01-01","1994"],"dc:description":["Combinatorial methods are used to prove several results in number theory. The chapters may be read independently, and are briefly discussed below.","In 1935 Erdos proved that every additive basis $\\rm{\\cal B}$ of order h is an essential component by establishing the inequality $\\rm \\sigma({\\cal A} + {\\cal B})\\ge \\sigma({\\cal A}) + {1\\over {2h}}\\sigma({\\cal A})(1-\\sigma({\\cal A})),$ where $\\rm\\sigma({\\cal A})$ denotes the Schnirelmann density of $\\rm{\\cal A}$. This lower bound was improved by Helmut Plunnecke in 1970 to $\\sigma({\\cal A} + {\\cal B})\\ge\\sigma ({\\cal A})\\sp{1-1/h}$ using an application of graph theory. A simplification of Plunnecke's proof is presented in Chapter 1.","The sequence of numbers $\\{ a\\sb{i}\\}$ defined by the recurrence $a\\sb{n} = (a\\sb{n-3}a\\sb{n-1} + a\\sbsp{n-2}{2})/a\\sb{n-4}$ for n $>$ 3, with initial values $a\\sb0, a\\sb1, a\\sb2, a\\sb3$ = 1, is shown to be integral in Chapter 2. The proof is extended to address more general sequences of this type.","In a famous work so entitled, Erdos and Selfridge established that the product of consecutive integers is never a power. In Chapter 3 related problems are considered in which one starts with n, not a kth power and selects a set of integers larger than n whose product with n forms a kth power, seeking to minimize the largest number used. In the restricted problem, the condition is placed on gaps between integers chosen so that no k consecutive numbers are omitted. In the case of squares, it is shown that the largest number used will not exceed 3n $-$ 3.","A set of integers is called sum-free if it contains no solution to the equation x + y = z. Erdos showed that every set of n integers has a sum-free subset with at least n/3 elements. This was strengthened by Alon and Kleitman to $>$n/3, and they showed by means of an example that the 1/3 cannot be improved to any number as large as 12/29(=.4137$\\...$). A construction is given in Chapter 4 which shows that it cannot be improved to 2/5.","A well-known theorem of Pillai and Szekeres states that for k $\\le$ 16, every set of k consecutive integers contains one which is relatively prime to the others. It was established by Brauer and Pillai that this is false for k $>$ 17. The condition of coprimality is strengthened to require that each of the numbers $\\{n,n + 1,\\..., n + k\\}$ have a factor in common with either n or n + k, and values of k for which this is possible are studied in Chapter 5.","Made available in DSpace on 2011-05-07T13:02:35Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512476.pdf: 1859542 bytes, checksum: cb9db6fc324ca7d79e98dc3352e8c2c9 (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:49:24Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:22:28-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9512476","(UMI)AAI9512476","http://hdl.handle.net/2142/21236"],"dc:language":["eng"],"dc:rights":["Copyright 1994 Malouf, Janice L."],"dc:subject":["Mathematics"],"dc:title":["Combinatorial approaches to integer sequences"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:17Z"}