{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T11:28:52Z","timestamp":1774610932149,"version":"3.50.1"},"reference-count":4,"publisher":"American Mathematical Society (AMS)","issue":"225","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>This paper presents some results concerning the search for initial values to the so-called <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 x plus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>3<\/mml:mn>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">3x+1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> problem which give rise either to function iterates that attain a maximum value higher than all function iterates for all smaller initial values, or which have a stopping time higher than those of all smaller initial values. Our computational results suggest that for an initial value of\u00a0<inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n  <mml:semantics>\n    <mml:mi>n<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, the maximum value of the function iterates is bounded from above by <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n squared f left-parenthesis n right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>n<\/mml:mi>\n        <mml:mn>2<\/mml:mn>\n      <\/mml:msup>\n      <mml:mi>f<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">n^2 f(n)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis n right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>f<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">f(n)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> either a constant or a very slowly increasing function of\u00a0<inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n  <mml:semantics>\n    <mml:mi>n<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. As a by-product of this (exhaustive) search, which was performed up to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 3 dot 2 Superscript 53 Baseline almost-equals 2.702 dot 10 Superscript 16\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>3<\/mml:mn>\n      <mml:mo>\u22c5<\/mml:mo>\n      <mml:msup>\n        <mml:mn>2<\/mml:mn>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>53<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo>\u2248<\/mml:mo>\n      <mml:mn>2.702<\/mml:mn>\n      <mml:mo>\u22c5<\/mml:mo>\n      <mml:msup>\n        <mml:mn>10<\/mml:mn>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>16<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">n=3 \\cdot 2^{53}\\approx 2.702 \\cdot 10^{16}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 x plus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>3<\/mml:mn>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">3x+1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> conjecture was verified up to that same number.<\/p>","DOI":"10.1090\/s0025-5718-99-01031-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:44Z","timestamp":1027721684000},"page":"371-384","source":"Crossref","is-referenced-by-count":29,"title":["Maximum excursion and stopping time record-holders for the problem: Computational results"],"prefix":"10.1090","volume":"68","author":[{"given":"Tom\u00e1s","family":"e Silva","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1999]]},"reference":[{"issue":"1-3","key":"1","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/0012-365X(93)90052-U","article-title":"The 3\ud835\udc65+1 problem: new lower bounds on nontrivial cycle lengths","volume":"118","author":"Eliahou, Shalom","year":"1993","journal-title":"Discrete Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-365X","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1214\/aoap\/1177005779","article-title":"The 3\ud835\udc65+1 problem: two stochastic models","volume":"2","author":"Lagarias, J. C.","year":"1992","journal-title":"Ann. Appl. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1050-5164","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"3","DOI":"10.2307\/2322189","article-title":"The 3\ud835\udc65+1 problem and its generalizations","volume":"92","author":"Lagarias, Jeffrey C.","year":"1985","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"11","key":"4","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0898-1221(92)90034-F","article-title":"3\ud835\udc65+1 search programs","volume":"24","author":"Leavens, Gary T.","year":"1992","journal-title":"Comput. Math. 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