{"id":"https://openalex.org/W7225810715","doi":"https://doi.org/10.5281/zenodo.24130044","title":"A power saving in the count of exceptions to Erd\u0151s's conjecture on ternary expansions of powers of 2","display_name":"A power saving in the count of exceptions to Erd\u0151s's conjecture on ternary expansions of powers of 2","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225810715","doi":"https://doi.org/10.5281/zenodo.24130044"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24130044","doi":"https://doi.org/10.5281/zenodo.24130044","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24130044","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24130044","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159949558","display_name":"Mario Heinrich Thaler","orcid":"https://orcid.org/0009-0001-6154-076X","observed_orcids":["https://orcid.org/0009-0001-6154-076X"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thaler, Mario Heinrich","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0001-6154-076X","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159949558"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0003000000142492354,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8381999731063843},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5435000061988831},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5425999760627747},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48170000314712524},{"id":"https://openalex.org/C6791720","wikidata":"https://www.wikidata.org/wiki/Q1670509","display_name":"Trinomial","level":2,"score":0.47200000286102295},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.435699999332428},{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.41830000281333923},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.3921000063419342},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.391400009393692},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.38760000467300415},{"id":"https://openalex.org/C188139237","wikidata":"https://www.wikidata.org/wiki/Q50706","display_name":"Square-free integer","level":2,"score":0.36410000920295715},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.359499990940094},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.35589998960494995},{"id":"https://openalex.org/C50742451","wikidata":"https://www.wikidata.org/wiki/Q1530388","display_name":"Equidistributed sequence","level":2,"score":0.337799996137619},{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.33219999074935913},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.3278999924659729},{"id":"https://openalex.org/C64452783","wikidata":"https://www.wikidata.org/wiki/Q1524945","display_name":"Ternary operation","level":2,"score":0.321399986743927},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.3098999857902527},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.30959999561309814},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.30709999799728394},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.302700012922287},{"id":"https://openalex.org/C2778738651","wikidata":"https://www.wikidata.org/wiki/Q16546687","display_name":"Novelty","level":2,"score":0.2732999920845032},{"id":"https://openalex.org/C93446932","wikidata":"https://www.wikidata.org/wiki/Q1972803","display_name":"Perfect power","level":2,"score":0.2651999890804291},{"id":"https://openalex.org/C2780286412","wikidata":"https://www.wikidata.org/wiki/Q1188344","display_name":"Unknot","level":3,"score":0.2572000026702881},{"id":"https://openalex.org/C143732363","wikidata":"https://www.wikidata.org/wiki/Q485520","display_name":"Goldbach's conjecture","level":3,"score":0.251800000667572}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24130044","doi":"https://doi.org/10.5281/zenodo.24130044","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24130044","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24130045","doi":"https://doi.org/10.5281/zenodo.24130045","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24130045","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24130044","doi":"https://doi.org/10.5281/zenodo.24130044","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24130044","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Erd\u0151s":[0],"conjectured":[1],"in":[2,111,128,232],"1979":[3],"that":[4,105,117],"for":[5,53,68,146,164,239],"every":[6,54],"n":[7,26],">":[8],"8":[9],"the":[10,16,22,65,80,83,95,106,121,137,147,160,199,203,216,226,228,243,262,265],"base-3":[11],"expansion":[12],"of":[13,24,82,114,192,218,254,256],"2^n":[14,71],"contains":[15],"digit":[17],"2.":[18],"Let":[19],"N(M)":[20,33,45],"be":[21],"number":[23],"exceptions":[25],"<":[27],"M.":[28,219],"The":[29,86,150,179,197],"classical":[30,84,161],"bound":[31,67],"is":[32,79,118,139,181,186,190],"\u226a":[34],"M^(log_3":[35],"2)":[36],"with":[37,58,72,97,131,142],"log_3":[38],"2":[39],"=":[40,60,62],"0.6309\u2026":[41],"This":[42,188],"note":[43,189],"proves":[44],"\u2264":[46],"(9/2)":[47],"\u00b7":[48,50,70,169,177],"M^\u03b2\u2080":[49],"log":[51],"M":[52,55,165],"\u2265":[56],"2,":[57],"\u03b2\u2080":[59],"log(9/5)/log(27/10)":[61],"0.5917\u2026,":[63],"and":[64,101,103,133,171,183,202,242],"same":[66],"\u03bb":[69],"any":[73],"nonzero":[74],"3-adic":[75],"integer":[76],"\u03bb,":[77],"which":[78],"generality":[81],"bound.":[85],"proof":[87],"bounds":[88],"Dirichlet":[89],"character":[90],"sums":[91],"modulo":[92],"3^k":[93],"over":[94],"integers":[96],"ternary":[98],"digits":[99],"0":[100],"1":[102],"deduces":[104],"admissible":[107],"exponents":[108],"are":[109,126,230],"equidistributed":[110],"initial":[112],"segments":[113],"a":[115],"period":[116],"longer":[119],"than":[120],"counting":[122],"range.":[123],"All":[124],"theorems":[125,229],"formalized":[127],"Lean":[129,200,233],"4":[130],"Mathlib":[132],"machine-checked":[134,231],"(Lean":[135],"4.35.0;":[136],"project":[138],"included,":[140],"together":[141],"exact":[143],"Python":[144],"computations":[145],"numerical":[148],"remarks).":[149],"result":[151],"does":[152],"not":[153,224],"answer":[154],"Erd\u0151s's":[155],"question.":[156],"It":[157],"improves":[158],"on":[159,261],"count":[162],"only":[163,173],"beyond":[166,174],"about":[167,175],"9":[168],"10^21,":[170],"throughout":[172],"3":[176],"10^27.":[178],"method":[180],"known,":[182],"no":[184],"novelty":[185],"claimed.":[187],"part":[191],"an":[193,208],"AI":[194,209],"publication":[195],"experiment.":[196],"mathematics,":[198],"proofs":[201],"text":[204],"were":[205],"written":[206],"by":[207],"system":[210],"(Claude":[211],"Fable":[212],"5,":[213],"Anthropic)":[214],"at":[215],"direction":[217],"H.":[220],"Thaler,":[221],"who":[222],"did":[223],"check":[225],"mathematics;":[227],"4.":[234],"Readers":[235],"should":[236],"consult":[237],"DETAILS.md":[238],"scope,":[240],"limitations":[241],"error":[244],"log.":[245],"For":[246,259],"background,":[247],"see":[248,264],"J.":[249],"C.":[250],"Lagarias,":[251],"\"Ternary":[252],"expansions":[253],"powers":[255],"2\"":[257],"(arXiv:math/0512006).":[258],"context":[260],"experiment,":[263],"retrospective":[266],"(DOI":[267],"10.5281/zenodo.17232247).":[268]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}