Bulletin of the Australian Mathematical Society最新文献

{"title":"ON A CONJECTURE ON SHIFTED PRIMES WITH LARGE PRIME FACTORS, II","authors":"YUCHEN DING","doi":"10.1017/s0004972724000534","DOIUrl":"https://doi.org/10.1017/s0004972724000534","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {P}$</span></span></img></span></span> be the set of primes and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$pi (x)$</span></span></img></span></span> the number of primes not exceeding <span>x</span>. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$P^+(n)$</span></span></img></span></span> be the largest prime factor of <span>n</span>, with the convention <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$P^+(1)=1$</span></span></img></span></span>, and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$ T_c(x)=#{ple x:pin mathcal {P},P^+(p-1)ge p^c}. $</span></span></img></span></span> Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, <span>Acta Math. Sin. (Engl. Ser.)</span> <span>33</span> (2017), 377–382], we show that for any <span>c</span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$8/9le c&lt;1$</span></span></img></span></span>, <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} limsup_{xrightarrowinfty}T_c(x)/pi(x)le 8(1/c-1), end{align*} $$</span></span></img></span></p><p>which clearly means that <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912124419965-0789:S0004972724000534:S0004972724000534_eqnu2.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} limsup_{xrightarrowinfty}T_c(x)/pi(x)rightarrow 0 quad text{as } crightarrow 1. end{align*} $$</span></span></img></span></p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MONOGENIC EVEN QUARTIC TRINOMIALS","authors":"LENNY JONES","doi":"10.1017/s0004972724000510","DOIUrl":"https://doi.org/10.1017/s0004972724000510","url":null,"abstract":"<p>A monic polynomial <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$f(x)in {mathbb Z}[x]$</span></span></img></span></span> of degree <span>N</span> is called <span>monogenic</span> if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$f(x)$</span></span></img></span></span> is irreducible over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb Q}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${1,theta ,theta ^2,ldots ,theta ^{N-1}}$</span></span></img></span></span> is a basis for the ring of integers of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb Q}(theta )$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$f(theta )=0$</span></span></img></span></span>. We prove that there exist exactly three distinct monogenic trinomials of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125559562-0272:S0004972724000510:S0004972724000510_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$x^4+bx^2+d$</span></span></img></span></span> whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ON A PROBLEM OF PONGSRIIAM ON THE SUM OF DIVISORS","authors":"RUI-JING WANG","doi":"10.1017/s0004972724000492","DOIUrl":"https://doi.org/10.1017/s0004972724000492","url":null,"abstract":"<p>For any positive integer <span>n</span>, let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913031000660-0696:S0004972724000492:S0004972724000492_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$sigma (n)$</span></span></img></span></span> be the sum of all positive divisors of <span>n</span>. We prove that for every integer <span>k</span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913031000660-0696:S0004972724000492:S0004972724000492_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$1leq kleq 29$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913031000660-0696:S0004972724000492:S0004972724000492_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(k,30)=1,$</span></span></img></span></span> <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913031000660-0696:S0004972724000492:S0004972724000492_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} sum_{nleq K}sigma(30n)&gt;sum_{nleq K}sigma(30n+k) end{align*} $$</span></span></img></span></p><p>for all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913031000660-0696:S0004972724000492:S0004972724000492_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Kin mathbb {N},$</span></span></img></span></span> which gives a positive answer to a problem posed by Pongsriiam [‘Sums of divisors on arithmetic progressions’, <span>Period. Math. Hungar</span>. <span>88</span> (2024), 443–460].</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"RADIAL ASYMPTOTICS OF GENERATING FUNCTIONS OF k-REGULAR SEQUENCES","authors":"MICHAEL COONS, JOHN LIND","doi":"10.1017/s0004972724000480","DOIUrl":"https://doi.org/10.1017/s0004972724000480","url":null,"abstract":"<p>We give a new proof of a theorem of Bell and Coons [‘Transcendence tests for Mahler functions’, <span>Proc. Amer. Math. Soc.</span> <span>145</span>(3) (2017), 1061–1070] on the leading order radial asymptotics of Mahler functions that are the generating functions of regular sequences. Our method allows us to provide a description of the oscillations whose existence was shown by Bell and Coons. This extends very recent results of Poulet and Rivoal [‘Radial behavior of Mahler functions’, <span>Int. J. Number Theory</span>, to appear].</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"EXTREMAL GRAPHS FOR DEGREE SUMS AND DOMINATING CYCLES","authors":"LU CHEN, YUEYU WU","doi":"10.1017/s0004972724000522","DOIUrl":"https://doi.org/10.1017/s0004972724000522","url":null,"abstract":"&lt;p&gt;A cycle &lt;span&gt;C&lt;/span&gt; of a graph &lt;span&gt;G&lt;/span&gt; is &lt;span&gt;dominating&lt;/span&gt; if &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$V(C)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is a dominating set and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$V(G)backslash V(C)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is an independent set. Wu &lt;span&gt;et al.&lt;/span&gt; [‘Degree sums and dominating cycles’, &lt;span&gt;Discrete Mathematics&lt;/span&gt; &lt;span&gt;344&lt;/span&gt; (2021), Article no. 112224] proved that every longest cycle of a &lt;span&gt;k&lt;/span&gt;-connected graph &lt;span&gt;G&lt;/span&gt; on &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$ngeq 3$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; vertices with &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$kgeq 2$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is dominating if the degree sum is more than &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$(k+1)(n+1)/3$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; for any &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$k+1$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs &lt;span&gt;G&lt;/span&gt; for this condition satisfy &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$(n-2)/3K_1vee (n+1)/3K_2 subseteq G subseteq K_{(n-2)/3}vee (n+1)/3K_2$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; or &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125250921-0666:S0004972724000522:S0004972724000522_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$2K_1vee 3K_{(n-2)/3}subseteq G subseteq K_2vee 3K_{(n-2)/3}.$&lt;/span&gt;&lt;/sp","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GRAPHS WITH SEMITOTAL DOMINATION NUMBER HALF THEIR ORDER","authors":"JIE CHEN, SHOU-JUN XU","doi":"10.1017/s0004972724000509","DOIUrl":"https://doi.org/10.1017/s0004972724000509","url":null,"abstract":"<p>In an isolate-free graph <span>G</span>, a subset <span>S</span> of vertices is a <span>semitotal dominating set</span> of <span>G</span> if it is a dominating set of <span>G</span> and every vertex in <span>S</span> is within distance 2 of another vertex of <span>S</span>. The <span>semitotal domination number</span> of <span>G</span>, denoted by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912125800045-0875:S0004972724000509:S0004972724000509_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$gamma _{t2}(G)$</span></span></img></span></span>, is the minimum cardinality of a semitotal dominating set in <span>G</span>. Goddard, Henning and McPillan [‘Semitotal domination in graphs’, <span>Utilitas Math.</span> <span>94</span> (2014), 67–81] characterised the trees and graphs of minimum degree 2 with semitotal domination number half their order. In this paper, we characterise all graphs whose semitotal domination number is half their order.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"INEQUALITIES AND UNIFORM ASYMPTOTIC FORMULAE FOR SPT-CRANK OF PARTITIONS","authors":"YUAN CHEN, NIAN HONG ZHOU","doi":"10.1017/s0004972724000455","DOIUrl":"https://doi.org/10.1017/s0004972724000455","url":null,"abstract":"<p>We establish some inequalities that arise from truncating Lerch sums and derive uniform asymptotic formulae for the spt-crank of ordinary partitions. The uniform asymptotic formulae improve upon a result of Mao [‘Asymptotic formulas for spt-crank of partitions’, <span>J. Math. Anal. Appl.</span> <span>460</span>(1) (2018), 121–139].</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A q-SUPERCONGRUENCE ARISING FROM ANDREWS’ IDENTITY","authors":"JI-CAI LIU, JING LIU","doi":"10.1017/s0004972724000467","DOIUrl":"https://doi.org/10.1017/s0004972724000467","url":null,"abstract":"We establish a <jats:italic>q</jats:italic>-analogue of a supercongruence related to a supercongruence of Rodriguez-Villegas, which extends a <jats:italic>q</jats:italic>-congruence of Guo and Zeng [‘Some <jats:italic>q</jats:italic>-analogues of supercongruences of Rodriguez-Villegas’, <jats:italic>J. Number Theory</jats:italic>145 (2014), 301–316]. The important ingredients in the proof include Andrews’ <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000467_inline2.png\"/> <jats:tex-math> $_4phi _3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> terminating identity.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MULTIPLICATIVE FUNCTIONS k-ADDITIVE ON GENERALISED OCTAGONAL NUMBERS","authors":"ELCHIN HASANALIZADE, POO-SUNG PARK","doi":"10.1017/s0004972724000479","DOIUrl":"https://doi.org/10.1017/s0004972724000479","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline1.png\"/&gt; &lt;jats:tex-math&gt; $kgeq 4$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be an integer. We prove that the set &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline2.png\"/&gt; &lt;jats:tex-math&gt; $mathcal {O}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of all nonzero generalised octagonal numbers is a &lt;jats:italic&gt;k&lt;/jats:italic&gt;-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline3.png\"/&gt; &lt;jats:tex-math&gt; $f_k$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; satisfies the condition &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_eqnu1.png\"/&gt; &lt;jats:tex-math&gt; $$ begin{align*} f_k(x_1+x_2+cdots+x_k)=f_k(x_1)+f_k(x_2)+cdots+f_k(x_k) end{align*} $$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt; for arbitrary &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline4.png\"/&gt; &lt;jats:tex-math&gt; $x_1,ldots ,x_kin mathcal {O}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, then &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline5.png\"/&gt; &lt;jats:tex-math&gt; $f_k$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is the identity function &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline6.png\"/&gt; &lt;jats:tex-math&gt; $f_k(n)=n$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; for all &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline7.png\"/&gt; &lt;jats:tex-math&gt; $nin mathbb {N}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. We also show that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline8.png\"/&gt; &lt;jats:tex-math&gt; $f_2$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000479_inline9.png\"/&gt; &lt;jats:tex-math&gt; $f_3$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; are not determined uniquel","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ENUMERATION OF GROUPS IN SOME SPECIAL VARIETIES OF A-GROUPS","authors":"ARUSHI, GEETHA VENKATARAMAN","doi":"10.1017/s0004972724000431","DOIUrl":"https://doi.org/10.1017/s0004972724000431","url":null,"abstract":"<p>We find an upper bound for the number of groups of order <span>n</span> up to isomorphism in the variety <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240824023521338-0591:S0004972724000431:S0004972724000431_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathfrak {S}}={mathfrak {A}_p}{mathfrak {A}_q}{mathfrak {A}_r}$</span></span></img></span></span>, where <span>p</span>, <span>q</span> and <span>r</span> are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240824023521338-0591:S0004972724000431:S0004972724000431_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathfrak {A}_qmathfrak {A}_r$</span></span></img></span></span>.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}