{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20414"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20414","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The representation of numbers as sums of unlike powers","abstract":"We are concerned with the problem of finding the least s for which every large natural number n admits a representation $n = x\\sbsp{2}{2} + x\\sbsp{3}{3} + \\cdots + x\\sbsp{s+1}{s+1}$, where the numbers $x\\sb{i}$ are nonnegative integers. K. F. Roth proved in 1948 that one may take s = 50, and this value has subsequently been reduced to s = 17 in a series of papers by K. Thanigasalam, R. C. Vaughan and J. Brudern. We improve this further, showing that s = 14 is admissible. This is accomplished by adapting a new iterative method, developed by Vaughan and T. D. Wooley for use in Waring's problem, for problems involving mixed powers. As with the other papers on the subject, the proof employs the machinery of the Hardy-Littlewood circle method. We also consider the problem of representing integers n in the form $n = x\\sbsp{k}{k} + x\\sbsp{k+1}{k+1} + \\cdots + x\\sbsp{k+s-1}{k+s-1}$, and obtain upper bounds on the number of terms required to represent every sufficiently large n in this form both for general k and for the specific case k = 3. The estimate obtained for general k improves an estimate by E. J. Scourfield. It is conjectured that in fact all large n can be written as the sum of a square, a positive cube and a fourth power of integers, and we give some numerical calculations that show that there are still many exceptions greater than 10$\\sp{18}$.","abstract_html":"We are concerned with the problem of finding the least s for which every large natural number n admits a representation $n = x\\sbsp{2}{2} + x\\sbsp{3}{3} + \\cdots + x\\sbsp{s+1}{s+1}$, where the numbers $x\\sb{i}$ are nonnegative integers. K. F. Roth proved in 1948 that one may take s = 50, and this value has subsequently been reduced to s = 17 in a series of papers by K. Thanigasalam, R. C. Vaughan and J. Brudern. We improve this further, showing that s = 14 is admissible. This is accomplished by adapting a new iterative method, developed by Vaughan and T. D. Wooley for use in Waring's problem, for problems involving mixed powers. As with the other papers on the subject, the proof employs the machinery of the Hardy-Littlewood circle method. We also consider the problem of representing integers n in the form $n = x\\sbsp{k}{k} + x\\sbsp{k+1}{k+1} + \\cdots + x\\sbsp{k+s-1}{k+s-1}$, and obtain upper bounds on the number of terms required to represent every sufficiently large n in this form both for general k and for the specific case k = 3. The estimate obtained for general k improves an estimate by E. J. Scourfield. It is conjectured that in fact all large n can be written as the sum of a square, a positive cube and a fourth power of integers, and we give some numerical calculations that show that there are still many exceptions greater than 10$\\sp{18}$.","abstract_has_math":true,"creators":["Ford, Kevin Barry"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:38:32Z","date_published":"2011-05-07T12:38:32Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Ford, Kevin Barry"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9503187","(UMI)AAI9503187"],"render_values":[{"text":"AAI9503187","href":null,"code":true},{"text":"(UMI)AAI9503187","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20414","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Ford, Kevin Barry"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:38:32Z","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 Ford, Kevin Barry"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9503187","(UMI)AAI9503187","http://hdl.handle.net/2142/20414"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We are concerned with the problem of finding the least s for which every large natural number n admits a representation $n = x\\sbsp{2}{2} + x\\sbsp{3}{3} + \\cdots + x\\sbsp{s+1}{s+1}$, where the numbers $x\\sb{i}$ are nonnegative integers. K. F. Roth proved in 1948 that one may take s = 50, and this value has subsequently been reduced to s = 17 in a series of papers by K. Thanigasalam, R. C. Vaughan and J. Brudern. We improve this further, showing that s = 14 is admissible. This is accomplished by adapting a new iterative method, developed by Vaughan and T. D. Wooley for use in Waring's problem, for problems involving mixed powers. As with the other papers on the subject, the proof employs the machinery of the Hardy-Littlewood circle method. We also consider the problem of representing integers n in the form $n = x\\sbsp{k}{k} + x\\sbsp{k+1}{k+1} + \\cdots + x\\sbsp{k+s-1}{k+s-1}$, and obtain upper bounds on the number of terms required to represent every sufficiently large n in this form both for general k and for the specific case k = 3. The estimate obtained for general k improves an estimate by E. J. Scourfield. It is conjectured that in fact all large n can be written as the sum of a square, a positive cube and a fourth power of integers, and we give some numerical calculations that show that there are still many exceptions greater than 10$\\sp{18}$.","Made available in DSpace on 2011-05-07T12:38:32Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9503187.pdf: 1687457 bytes, checksum: 1bb92a21a90b63267eaf3d9da152756f (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:44Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:10-05:00 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":["The representation of numbers as sums of unlike powers"]}]}],"canonical_facts":{"dc:creator":["Ford, Kevin Barry"],"dc:date":["2011-05-07T12:38:32Z","10000-01-01","1994"],"dc:description":["We are concerned with the problem of finding the least s for which every large natural number n admits a representation $n = x\\sbsp{2}{2} + x\\sbsp{3}{3} + \\cdots + x\\sbsp{s+1}{s+1}$, where the numbers $x\\sb{i}$ are nonnegative integers. K. F. Roth proved in 1948 that one may take s = 50, and this value has subsequently been reduced to s = 17 in a series of papers by K. Thanigasalam, R. C. Vaughan and J. Brudern. We improve this further, showing that s = 14 is admissible. This is accomplished by adapting a new iterative method, developed by Vaughan and T. D. Wooley for use in Waring's problem, for problems involving mixed powers. As with the other papers on the subject, the proof employs the machinery of the Hardy-Littlewood circle method. We also consider the problem of representing integers n in the form $n = x\\sbsp{k}{k} + x\\sbsp{k+1}{k+1} + \\cdots + x\\sbsp{k+s-1}{k+s-1}$, and obtain upper bounds on the number of terms required to represent every sufficiently large n in this form both for general k and for the specific case k = 3. The estimate obtained for general k improves an estimate by E. J. Scourfield. It is conjectured that in fact all large n can be written as the sum of a square, a positive cube and a fourth power of integers, and we give some numerical calculations that show that there are still many exceptions greater than 10$\\sp{18}$.","Made available in DSpace on 2011-05-07T12:38:32Z (GMT). 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