Pub Date : 2026-02-03DOI: 10.1016/j.jnt.2025.12.012
Wu-Xia Ma , Yong-Gao Chen
Text
Let W be a non-empty subset of . A set is called an additive complement to W if . An additive complement C is said to be minimal if no proper subset of C is an additive complement to W. In 2019, Kiss, Sándor, and Yang proved that if m is a positive integer and , where X and Y are finite sets that satisfy certain conditions, then W has a minimal additive complement. They asked whether this is true if Y is an infinite set and does not contain an arithmetic progression of common difference m. In this paper, we present some results on this problem and pose three open problems for further research.
Video
For a video summary of this paper, please visit https://youtu.be/eSZCqS99au0.
设W为Z的一个非空子集。当集C+W=Z时，称集C≠Z为W的加性补。在2019年，Kiss, Sándor， and Yang证明了如果m是正整数且W=(mN+X)∪Y，其中X和Y是满足一定条件的有限集，则W具有最小的可加补。他们问，如果Y是一个无限集，并且不包含公差m的等差数列，这是否成立。在本文中，我们给出了这个问题的一些结果，并提出了三个开放的问题供进一步研究。观看本文的视频摘要，请访问https://youtu.be/eSZCqS99au0。
{"title":"On a problem of minimal additive complements of integers","authors":"Wu-Xia Ma , Yong-Gao Chen","doi":"10.1016/j.jnt.2025.12.012","DOIUrl":"10.1016/j.jnt.2025.12.012","url":null,"abstract":"<div><h3>Text</h3><div>Let <em>W</em> be a non-empty subset of <span><math><mi>Z</mi></math></span>. A set <span><math><mi>C</mi><mo>⊆</mo><mi>Z</mi></math></span> is called an additive complement to <em>W</em> if <span><math><mi>C</mi><mo>+</mo><mi>W</mi><mo>=</mo><mi>Z</mi></math></span>. An additive complement <em>C</em> is said to be minimal if no proper subset of <em>C</em> is an additive complement to <em>W</em>. In 2019, Kiss, Sándor, and Yang proved that if <em>m</em> is a positive integer and <span><math><mi>W</mi><mo>=</mo><mo>(</mo><mi>m</mi><mi>N</mi><mo>+</mo><mi>X</mi><mo>)</mo><mo>∪</mo><mi>Y</mi></math></span>, where <em>X</em> and <em>Y</em> are finite sets that satisfy certain conditions, then <em>W</em> has a minimal additive complement. They asked whether this is true if <em>Y</em> is an infinite set and does not contain an arithmetic progression of common difference <em>m</em>. In this paper, we present some results on this problem and pose three open problems for further research.</div></div><div><h3>Video</h3><div>For a video summary of this paper, please visit <span><span>https://youtu.be/eSZCqS99au0</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 178-187"},"PeriodicalIF":0.7,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-21DOI: 10.1016/j.jnt.2025.12.005
Bruno Aguiló-Vidal , Luis Arenas-Carmona , Matías Saavedra-Lagos
We apply the theory of branches in Bruhat-Tits trees, developed in previous works by the second author and others, to the study of two dimensional representations of finite groups over the ring of integers of a number field. We provide a general strategy to perform these computations, and we give explicit formulas for some particular families.
{"title":"Two dimensional integral representations via branches of the Bruhat-Tits tree","authors":"Bruno Aguiló-Vidal , Luis Arenas-Carmona , Matías Saavedra-Lagos","doi":"10.1016/j.jnt.2025.12.005","DOIUrl":"10.1016/j.jnt.2025.12.005","url":null,"abstract":"<div><div>We apply the theory of branches in Bruhat-Tits trees, developed in previous works by the second author and others, to the study of two dimensional representations of finite groups over the ring of integers of a number field. We provide a general strategy to perform these computations, and we give explicit formulas for some particular families.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 62-103"},"PeriodicalIF":0.7,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-21DOI: 10.1016/j.jnt.2025.12.008
Aidan Botkin , Madeline L. Dawsey , David J. Hemmer , Matthew R. Just , Robert Schneider
We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for , where is the kth prime number, is the divisor function, and is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of similar to those provided by the logarithmic integral, and much more accurate than up to where the estimates are almost exact. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.
{"title":"Partition-theoretic model of prime distribution","authors":"Aidan Botkin , Madeline L. Dawsey , David J. Hemmer , Matthew R. Just , Robert Schneider","doi":"10.1016/j.jnt.2025.12.008","DOIUrl":"10.1016/j.jnt.2025.12.008","url":null,"abstract":"<div><div>We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>,<span><span><span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mo>=</mo><mspace></mspace><mn>1</mn><mspace></mspace><mo>+</mo><mspace></mspace><mn>2</mn><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mrow><mo>⌈</mo><mfrac><mrow><mi>d</mi><mo>(</mo><mi>j</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mspace></mspace><mo>+</mo><mspace></mspace><mi>ε</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the <em>k</em>th prime number, <span><math><mi>d</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> is the divisor function, and <span><math><mi>ε</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of <span><math><mi>π</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> similar to those provided by the logarithmic integral, and much more accurate than <span><math><mi>li</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> up to <span><math><mi>n</mi><mo>=</mo><mn>10</mn><mo>,</mo><mn>000</mn></math></span> where the estimates are <em>almost exact</em>. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 104-130"},"PeriodicalIF":0.7,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a finite abelian group and k be an integer not dividing the exponent of G. We denote by the smallest positive integer l such that every sequence over G of length no less than l has a zero-sum subsequence of length not divisible by k. In this paper, we focus on determining for , a cyclic group of order n. Specifically, we prove that for .
{"title":"On the existence of zero-sum subsequences with length not divided by a given number","authors":"Weidong Gao , Xiao Jiang , Yuanlin Li , Huijuan Qi","doi":"10.1016/j.jnt.2025.12.002","DOIUrl":"10.1016/j.jnt.2025.12.002","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group and <em>k</em> be an integer not dividing the exponent of <em>G</em>. We denote by <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> the smallest positive integer <em>l</em> such that every sequence over <em>G</em> of length no less than <em>l</em> has a zero-sum subsequence of length not divisible by <em>k</em>. In this paper, we focus on determining <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, a cyclic group of order <em>n</em>. Specifically, we prove that<span><span><span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>⌊</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span></span></span> for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>}</mo><mo>∪</mo><mo>(</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 15-37"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.003
Ethan Simpson Lee , Paweł Nosal
In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that and are bounded from above by for all and respectively, where and are the Chebyshev ψ and ϑ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.
{"title":"Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis","authors":"Ethan Simpson Lee , Paweł Nosal","doi":"10.1016/j.jnt.2025.12.003","DOIUrl":"10.1016/j.jnt.2025.12.003","url":null,"abstract":"<div><div>In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that <span><math><mo>|</mo><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>x</mi><mo>|</mo></math></span> and <span><math><mo>|</mo><mi>ϑ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>x</mi><mo>|</mo></math></span> are bounded from above by<span><span><span><math><mfrac><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt><mi>log</mi><mo></mo><mi>x</mi><mo>(</mo><mi>log</mi><mo></mo><mi>x</mi><mo>−</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>8</mn><mi>π</mi></mrow></mfrac></math></span></span></span> for all <span><math><mi>x</mi><mo>≥</mo><mn>101</mn></math></span> and <span><math><mi>x</mi><mo>≥</mo><mn>2</mn><mspace></mspace><mn>657</mn></math></span> respectively, where <span><math><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>ϑ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are the Chebyshev <em>ψ</em> and <em>ϑ</em> functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 241-258"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.001
Seokhyun Choi, Bo-Hae Im
We prove Larsen's conjecture for elliptic curves over with analytic rank at most 1. Specifically, let be an elliptic curve over . If has analytic rank at most 1, then we prove that for any topologically finitely generated subgroup G of , the rank of E over the fixed subfield of under G is infinite.
{"title":"Larsen's conjecture for elliptic curves over Q with analytic rank at most one","authors":"Seokhyun Choi, Bo-Hae Im","doi":"10.1016/j.jnt.2025.12.001","DOIUrl":"10.1016/j.jnt.2025.12.001","url":null,"abstract":"<div><div>We prove Larsen's conjecture for elliptic curves over <span><math><mi>Q</mi></math></span> with analytic rank at most 1. Specifically, let <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> be an elliptic curve over <span><math><mi>Q</mi></math></span>. If <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> has analytic rank at most 1, then we prove that for any topologically finitely generated subgroup <em>G</em> of <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover><mo>/</mo><mi>Q</mi><mo>)</mo></math></span>, the rank of <em>E</em> over the fixed subfield <span><math><msup><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msup></math></span> of <span><math><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></math></span> under <em>G</em> is infinite.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 1-14"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.004
Thomas H. Geisser
We give a version of the Artin-Tate formula for surfaces over a finite field which holds without assuming Tate's conjecture for divisors. The formula gives an equality between terms related to the Brauer group on the one hand, and terms related to the Néron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to give sharp estimates on the size of the Brauer group of abelian surfaces depending on the p-rank.
{"title":"Brauer and Néron-Severi groups of surfaces over finite fields","authors":"Thomas H. Geisser","doi":"10.1016/j.jnt.2025.12.004","DOIUrl":"10.1016/j.jnt.2025.12.004","url":null,"abstract":"<div><div>We give a version of the Artin-Tate formula for surfaces over a finite field which holds without assuming Tate's conjecture for divisors. The formula gives an equality between terms related to the Brauer group on the one hand, and terms related to the Néron-Severi group on the other hand. We give estimates on the terms appearing in the formula and use this to give sharp estimates on the size of the Brauer group of abelian surfaces depending on the <em>p</em>-rank.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 38-61"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.006
Biao Wang , Shaoyun Yi
Let be an irreducible polynomial of degree . Let be an integer. The number of integers n such that is k-free is widely studied in the literature. In principle, one expects that is k-free infinitely often, if f has no fixed k-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.
{"title":"The prime number theorem over integers of power-free polynomial values","authors":"Biao Wang , Shaoyun Yi","doi":"10.1016/j.jnt.2025.12.006","DOIUrl":"10.1016/j.jnt.2025.12.006","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> be an irreducible polynomial of degree <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span>. Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer. The number of integers <em>n</em> such that <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is <em>k</em>-free is widely studied in the literature. In principle, one expects that <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is <em>k</em>-free infinitely often, if <em>f</em> has no fixed <em>k</em>-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 216-229"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-19DOI: 10.1016/j.jnt.2025.12.007
Kegang Liu , Zicheng Qian
We prove a mod-p local-global compatibility result for Scholze's functor in higher dimensions, under certain multiplicity-free condition. This improves the previous result in this direction of K. Liu, by removing the semisimple assumption on the mod p Galois representations. Our proof relies mainly on a criterion for σ-typicity of modules which is obtained by representation-theoretic techniques.
{"title":"A note on mod-p local-global compatibility via Scholze's functor","authors":"Kegang Liu , Zicheng Qian","doi":"10.1016/j.jnt.2025.12.007","DOIUrl":"10.1016/j.jnt.2025.12.007","url":null,"abstract":"<div><div>We prove a mod-<em>p</em> local-global compatibility result for Scholze's functor in higher dimensions, under certain multiplicity-free condition. This improves the previous result in this direction of K. Liu, by removing the semisimple assumption on the mod <em>p</em> Galois representations. Our proof relies mainly on a criterion for <em>σ</em>-typicity of modules which is obtained by representation-theoretic techniques.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 230-240"},"PeriodicalIF":0.7,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-30DOI: 10.1016/j.jnt.2025.11.007
Pedro-José Cazorla García
Given a prime number q and a squarefree integer , we develop a method to explicitly determine the tuples for which the difference has squarefree part equal to . Our techniques include the combination of the local information provided by Galois representations of Frey–Hellegouarch curves with the effective resolution of Thue–Mahler equations, as well as the use of improved lower bounds for q-adic and complex logarithms. As an application of this methodology, we will completely resolve the case when and .
{"title":"On differences of perfect powers and prime powers","authors":"Pedro-José Cazorla García","doi":"10.1016/j.jnt.2025.11.007","DOIUrl":"10.1016/j.jnt.2025.11.007","url":null,"abstract":"<div><div>Given a prime number <em>q</em> and a squarefree integer <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, we develop a method to explicitly determine the tuples <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> for which the difference <span><math><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span> has squarefree part equal to <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. Our techniques include the combination of the local information provided by Galois representations of Frey–Hellegouarch curves with the effective resolution of Thue–Mahler equations, as well as the use of improved lower bounds for <em>q</em>-adic and complex logarithms. As an application of this methodology, we will completely resolve the case when <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mn>20</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>q</mi><mo><</mo><mn>25</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 121-169"},"PeriodicalIF":0.7,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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