Bulletin of the Australian Mathematical Society最新文献

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ON A CONJECTURE ON SHIFTED PRIMES WITH LARGE PRIME FACTORS, II 关于大质因数移位素数的猜想，ii

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000534

YUCHEN DING

Let $mathcal {P}$ be the set of primes and $pi (x)$ the number of primes not exceeding x. Let $P^+(n)$ be the largest prime factor of n, with the convention $P^+(1)=1$, and $ T_c(x)=#{ple x:pin mathcal {P},P^+(p-1)ge p^c}. $ Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.) 33 (2017), 377–382], we show that for any c with $8/9le c<1$, $$ begin{align*} limsup_{xrightarrowinfty}T_c(x)/pi(x)le 8(1/c-1), end{align*} $$

which clearly means that $$ begin{align*} limsup_{xrightarrowinfty}T_c(x)/pi(x)rightarrow 0 quad text{as } crightarrow 1. end{align*} $$

让 $P^+(n)$ 是 n 的最大素因子，约定为 $P^+(1)=1$，并且 $ T_c(x)=#{ple x:pin mathcal {P},P^+(p-1)ge p^c}.$ 由陈和陈的一个猜想激发['论移位素数的最大素因子', Acta Math.Sin.(Engl. Ser.) 33 (2017), 377-382], 我们证明，对于任意具有 $8/9le c<1$ 的 c，$$ begin{align*}limsup_{xrightarrowinfty}T_c(x)/pi(x)le 8(1/c-1), end{align*}这显然意味着 $$ （开始{align*}T_c(x)/pi(x)rightarrow 0 quad text{as } crightarrow 1.end{align*}$$

引用次数: 0

MONOGENIC EVEN QUARTIC TRINOMIALS 单向偶四次方三项式

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000510

LENNY JONES

A monic polynomial $f(x)in {mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${mathbb Q}$ and ${1,theta ,theta ^2,ldots ,theta ^{N-1}}$ is a basis for the ring of integers of ${mathbb Q}(theta )$, where $f(theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ 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.

如果 $f(x)$ 在 ${mathbb Q}$ 上是不可约的，并且 ${1、theta ,theta ^2,ldots ,theta ^{N-1}}$ 是 ${mathbb Q}(theta )$ 的整数环的基，其中 $f(theta )=0$.我们证明恰好存在三个不同的形式为 $x^4+bx^2+d$ 的单元三项式，它们的伽罗瓦群是阶数为 4 的循环群。我们还证明了当伽罗瓦群不是循环群时，情况会截然不同。

引用次数: 0

ON A PROBLEM OF PONGSRIIAM ON THE SUM OF DIVISORS 关于除数之和的庞氏难题

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000492

RUI-JING WANG

For any positive integer n, let $sigma (n)$ be the sum of all positive divisors of n. We prove that for every integer k with $1leq kleq 29$ and $(k,30)=1,$ $$ begin{align*} sum_{nleq K}sigma(30n)>sum_{nleq K}sigma(30n+k) end{align*} $$

for all $Kin mathbb {N},$ which gives a positive answer to a problem posed by Pongsriiam [‘Sums of divisors on arithmetic progressions’, Period. Math. Hungar. 88 (2024), 443–460].

对于任意正整数 n，让 $sigma (n)$ 是 n 的所有正除数之和。我们证明，对于每一个整数k，只要有$1leq kleq 29$和$(k,30)=1,$$ begin{align*}。sum_{nleq K}sigma(30n)>sum_{nleq K}sigma(30n+k) end{align*}$$for all $Kin mathbb {N}, $ 这给了 Pongsriiam 提出的问题一个肯定的答案['Sums of divisors on arithmetic progressions', Period.Math.匈牙利。88 (2024), 443-460].

引用次数: 0

RADIAL ASYMPTOTICS OF GENERATING FUNCTIONS OF k-REGULAR SEQUENCES k 规则序列生成函数的径向拟数

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000480

MICHAEL COONS, JOHN LIND

We give a new proof of a theorem of Bell and Coons [‘Transcendence tests for Mahler functions’, Proc. Amer. Math. Soc. 145(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’, Int. J. Number Theory, to appear].

我们给出了贝尔和库恩斯定理的新证明['马勒函数的超越性检验'，Proc.Amer.Math.145(3)（2017），1061-1070]关于作为正则序列生成函数的马勒函数的前阶径向渐近性的新证明。我们的方法允许我们对贝尔和库恩斯证明存在的振荡进行描述。这扩展了 Poulet 和 Rivoal 的最新成果['马勒函数的径向行为'，《国际数论杂志》，待出版]。

引用次数: 0

EXTREMAL GRAPHS FOR DEGREE SUMS AND DOMINATING CYCLES 度数和主循环的极值图

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000522

LU CHEN, YUEYU WU

A cycle C of a graph G is dominating if <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">$V(C)$</img> is a dominating set and <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">$V(G)backslash V(C)$</img> is an independent set. Wu et al. [‘Degree sums and dominating cycles’, Discrete Mathematics 344 (2021), Article no. 112224] proved that every longest cycle of a k-connected graph G on <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">$ngeq 3$</img> vertices with <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">$kgeq 2$</img> is dominating if the degree sum is more than <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">$(k+1)(n+1)/3$</img> for any <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">$k+1$</img> pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs G for this condition satisfy <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">$(n-2)/3K_1vee (n+1)/3K_2 subseteq G subseteq K_{(n-2)/3}vee (n+1)/3K_2$</img> or <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">$2K_1vee 3K_{(n-2)/3}subseteq G subseteq K_2vee 3K_{(n-2)/3}.$</sp

如果 $V(C)$ 是支配集且 $V(G)backslash V(C)$ 是独立集，那么图 G 的循环 C 就是支配集。Wu 等人['度和与支配循环'，《离散数学》344 (2021)，文章编号：112224]证明了每一个最长的循环都是支配循环。112224]证明了如果对于任意 $k+1$ 成对非相邻顶点的度数总和大于 $(k+1)(n+1)/3$，则在 $ngeq 3$ 顶点上具有 $kgeq 2$ 的 k 连接图 G 的每个最长循环都是支配循环。他们还证明了这一界限是尖锐的。在本文中，我们证明了这个条件下的极值图 G 满足 $(n-2)/3K_1vee (n+1)/3K_2 subseteq G subseteq K_{(n-2)/3}vee (n+1)/3K_2$ 或 $2K_1vee 3K_{(n-2)/3}subseteq G subseteq K_2vee 3K_{(n-2)/3}.$ 。

引用次数: 0

GRAPHS WITH SEMITOTAL DOMINATION NUMBER HALF THEIR ORDER 半总支配数为其阶数一半的图

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000509

JIE CHEN, SHOU-JUN XU

In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by $gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Goddard, Henning and McPillan [‘Semitotal domination in graphs’, Utilitas Math. 94 (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.

在无孤立图 G 中，如果顶点子集 S 是 G 的支配集，且 S 中的每个顶点与 S 中另一个顶点的距离都在 2 以内，则该顶点子集 S 是 G 的半总支配集。G 的半总支配数用 $gamma _{t2}(G)$ 表示，是 G 中半总支配集的最小心性。Goddard、Henning 和 McPillan ['图中的半总支配数'，Utilitas Math.在本文中，我们将描述所有半总支配数为其阶数一半的图的特征。

引用次数: 0

INEQUALITIES AND UNIFORM ASYMPTOTIC FORMULAE FOR SPT-CRANK OF PARTITIONS 分区的不等式和均匀渐近公式

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-09-13 DOI: 10.1017/s0004972724000455

YUAN CHEN, NIAN HONG ZHOU

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’, J. Math. Anal. Appl. 460(1) (2018), 121–139].

我们建立了截断勒奇和所产生的一些不等式，并推导出了普通分区 spt-rank的统一渐近公式。统一渐近公式改进了毛泽东的一个结果['Asymptotic formulas for spt-crank of partitions', J. Math.Anal.460(1) (2018)，121-139]。

引用次数: 0

A q-SUPERCONGRUENCE ARISING FROM ANDREWS’ IDENTITY 由安德鲁斯的身份产生的 q 级超不确定性

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-08-29 DOI: 10.1017/s0004972724000467

JI-CAI LIU, JING LIU

We establish a q-analogue of a supercongruence related to a supercongruence of Rodriguez-Villegas, which extends a q-congruence of Guo and Zeng [‘Some q-analogues of supercongruences of Rodriguez-Villegas’, J. Number Theory145 (2014), 301–316]. The important ingredients in the proof include Andrews’

$_4phi _3$

terminating identity.

我们建立了一个与罗德里格斯-维耶加斯的超等公差相关的 q-analogue ，它扩展了郭和曾的 q-ongruence ['罗德里格斯-维耶加斯的超等公差的一些 q-analogue', J. Number Theory145 (2014), 301-316] 。证明中的重要成分包括安德鲁斯的$_4phi _3$终止身份。

引用次数: 0

MULTIPLICATIVE FUNCTIONS k-ADDITIVE ON GENERALISED OCTAGONAL NUMBERS 公有四舍五入数的多元函数 k- ADDITIVE

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-08-27 DOI: 10.1017/s0004972724000479

ELCHIN HASANALIZADE, POO-SUNG PARK

Let $kgeq 4$ be an integer. We prove that the set $mathcal {O}$ of all nonzero generalised octagonal numbers is a k-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function $f_k$ satisfies the condition $$ begin{align*} f_k(x_1+x_2+cdots+x_k)=f_k(x_1)+f_k(x_2)+cdots+f_k(x_k) end{align*} $$ for arbitrary $x_1,ldots ,x_kin mathcal {O}$ , then $f_k$ is the identity function $f_k(n)=n$ for all $nin mathbb {N}$ . We also show that $f_2$ and $f_3$ are not determined uniquel

让 $kgeq 4$ 是一个整数。我们证明所有非零广义八角数的集合 $mathcal {O}$ 是乘法函数集合的 k-additive uniqueness 集合。也就是说，如果一个乘法函数 $f_k$ 满足条件 $$ begin{align*} f_k(x_1+x_2+cdots+x_k)=f_k(x_1)+f_k(x_2)+cdots+f_k(x_k) end{align*}$$ for arbitrary $x_1,ldots ,x_kin mathcal {O}$ , then $f_k$ is the identity function $f_k(n)=n$ for all $nin mathbb {N}$.我们还证明 $f_2$ 和 $f_3$ 并不是唯一确定的。

引用次数: 0

ENUMERATION OF GROUPS IN SOME SPECIAL VARIETIES OF A-GROUPS A 群的一些特殊品种中的群的枚举

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-08-27 DOI: 10.1017/s0004972724000431

ARUSHI, GEETHA VENKATARAMAN

We find an upper bound for the number of groups of order n up to isomorphism in the variety ${mathfrak {S}}={mathfrak {A}_p}{mathfrak {A}_q}{mathfrak {A}_r}$, where p, q and r 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 $mathfrak {A}_qmathfrak {A}_r$.

我们发现了在 ${mathfrak {S}}={mathfrak {A}_p}{mathfrak {A}_q}{mathfrak {A}_r}$ 变项（其中 p、q 和 r 是不同的素数）中阶数为 n 的同构群的数量上限。我们还找到了在一般线性群的子群中也在 $mathfrak {A}_qmathfrak {A}_r}$ 中的最大子群的阶数和共轭类数的约束。

引用次数: 0

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