{"title":"A BBP-style computation for $π$ in base 10","authors":"Wadim Zudilin","doi":"arxiv-2409.10097","DOIUrl":"https://doi.org/arxiv-2409.10097","url":null,"abstract":"We articulate about how to compute (promptly) the digits of $pi$, in bases 5\u0000and 10, from a given place without computing preceding ones.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ehud de ShalitUT3, IMT, IUF, Charlotte HardouinUT3, IMT, IUF, Julien RoquesICJ, CTN
The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $phi f(z)=f(qz)$, $2le qinmathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},zeta(z,Lambda)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $zeta$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch''afke and Singer over the projective line.
{"title":"Hypertranscendence and $q$-difference equations over elliptic functionfields","authors":"Ehud de ShalitUT3, IMT, IUF, Charlotte HardouinUT3, IMT, IUF, Julien RoquesICJ, CTN","doi":"arxiv-2409.10092","DOIUrl":"https://doi.org/arxiv-2409.10092","url":null,"abstract":"The differential nature of solutions of linear difference equations over the\u0000projective line was recently elucidated. In contrast, little is known about the\u0000differential nature of solutions of linear difference equations over elliptic\u0000curves. In the present paper, we study power series $f(z)$ with complex\u0000coefficients satisfying a linear difference equation over a field of elliptic\u0000functions $K$,with respect to the difference operator $phi f(z)=f(qz)$, $2le\u0000qinmathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main\u0000theoremsays that such an $f$ satisfies, in addition, a polynomial\u0000differentialequation with coefficients from $K,$ if and only if it belongs\u0000tothe ring $S=K[z,z^{-1},zeta(z,Lambda)]$ generated over $K$ by$z,z^{-1}$ and\u0000the Weierstrass $zeta$-function. This is the first elliptic extension of\u0000recent theorems of Adamczewski, Dreyfus and Hardouin concerning the\u0000differential transcendence of solutions of difference equations with\u0000coefficients in $mathbb{C}(z),$ in which various difference operators were\u0000considered (shifts, $q$-differenceoperators or Mahler operators). While the\u0000general approach, of usingparametrized Picard-Vessiot theory, is similar, many\u0000features, andin particular the emergence of monodromy considerations and the\u0000ring$S$, are unique to the elliptic case and are responsible for non-trivial\u0000difficulties. We emphasize that, among the intermediate results, we prove an\u0000integrability result for difference-differential systems over ellipticcurves\u0000which is a genus one analogue of the integrability results obtained by\u0000Sch''afke and Singer over the projective line.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $omega(n)$ denote the number of distinct prime factors of a natural number $n$. In 1917, Hardy and Ramanujan proved that $omega(n)$ has normal order $log log n$ over naturals. In this work, we establish the first and the second moments of $omega(n)$ over $h$-free and $h$-full numbers using a new counting argument and prove that $omega(n)$ has normal order $log log n$ over these subsets.
{"title":"Distribution of $ω(n)$ over $h$-free and $h$-full numbers","authors":"Sourabhashis Das, Wentang Kuo, Yu-Ru Liu","doi":"arxiv-2409.10430","DOIUrl":"https://doi.org/arxiv-2409.10430","url":null,"abstract":"Let $omega(n)$ denote the number of distinct prime factors of a natural\u0000number $n$. In 1917, Hardy and Ramanujan proved that $omega(n)$ has normal\u0000order $log log n$ over naturals. In this work, we establish the first and the\u0000second moments of $omega(n)$ over $h$-free and $h$-full numbers using a new\u0000counting argument and prove that $omega(n)$ has normal order $log log n$\u0000over these subsets.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"214 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Osama Salah, A. Elsonbaty, Mohammed Abdul Azim Seoud, Mohamed Anwar
This paper finds all Lucas numbers which are the sum of two Jacobsthal numbers. It also finds all Jacobsthal numbers which are the sum of two Lucas numbers. In general, we find all non-negative integer solutions $(n, m, k)$ of the two Diophantine equations $L_n +L_m =J_k$ and $J_n +J_m =L_K,$ where $leftlbrace L_{k}rightrbrace_{kgeq0}$ and $leftlbrace J_{n}rightrbrace_{ngeq0}$ are the sequences of Lucas and Jacobsthal numbers, respectively. Our primary results are supported by an adaption of the Baker's theorem for linear forms in logarithms and Dujella and PethH{o}'s reduction method.
{"title":"On the Diophantine Equations $J_n +J_m =L_k$ and $L_n +L_m =J_k$","authors":"Osama Salah, A. Elsonbaty, Mohammed Abdul Azim Seoud, Mohamed Anwar","doi":"arxiv-2409.09791","DOIUrl":"https://doi.org/arxiv-2409.09791","url":null,"abstract":"This paper finds all Lucas numbers which are the sum of two Jacobsthal\u0000numbers. It also finds all Jacobsthal numbers which are the sum of two Lucas\u0000numbers. In general, we find all non-negative integer solutions $(n, m, k)$ of\u0000the two Diophantine equations $L_n +L_m =J_k$ and $J_n +J_m =L_K,$ where\u0000$leftlbrace L_{k}rightrbrace_{kgeq0}$ and $leftlbrace\u0000J_{n}rightrbrace_{ngeq0}$ are the sequences of Lucas and Jacobsthal numbers,\u0000respectively. Our primary results are supported by an adaption of the Baker's\u0000theorem for linear forms in logarithms and Dujella and PethH{o}'s reduction\u0000method.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
N. Bradley Fox, Nathan H. Fox, Helen G. Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot
For a base $b geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the number of iterations required to map it to $1$. We determine the fixed points and cycles of $E_{2,b}$ and prove a range of results concerning sequences of $b$-elated numbers and $b$-elated numbers of minimal heights. Although the $b$-elated function is closely related to the $b$-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work.
{"title":"Elated Numbers","authors":"N. Bradley Fox, Nathan H. Fox, Helen G. Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot","doi":"arxiv-2409.09863","DOIUrl":"https://doi.org/arxiv-2409.09863","url":null,"abstract":"For a base $b geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive\u0000integer written in base $b$ to the product of its leading digit and the sum of\u0000the squares of its digits. A $b$-elated number is a positive integer that maps\u0000to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the\u0000number of iterations required to map it to $1$. We determine the fixed points\u0000and cycles of $E_{2,b}$ and prove a range of results concerning sequences of\u0000$b$-elated numbers and $b$-elated numbers of minimal heights. Although the\u0000$b$-elated function is closely related to the $b$-happy function, the behaviors\u0000of the two are notably different, as demonstrated by the results in this work.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss certain matrices associated with Christoffel words, and show that they have a group structure. We compute their determinants and show a relationship between the Zolotareff symbol from number theory.
{"title":"Christoffel Matrices and Sturmian Determinants","authors":"Christophe Reutenauer, Jeffrey Shallit","doi":"arxiv-2409.09824","DOIUrl":"https://doi.org/arxiv-2409.09824","url":null,"abstract":"We discuss certain matrices associated with Christoffel words, and show that\u0000they have a group structure. We compute their determinants and show a\u0000relationship between the Zolotareff symbol from number theory.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explicitly describe the splitting of odd integral primes in the radical extension $mathbb{Q}(sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $mathbb{Z}[x]$. Our motivation is to classify common index divisors, the primes whose splitting prevents the existence of a power integral basis for the ring of integers of $mathbb{Q}(sqrt[n]{a})$. Among other results, we show that if $p$ is such a prime, even or otherwise, then $pmid n$.
{"title":"Prime Splitting and Common Index Divisors in Radical Extensions","authors":"Hanson Smith","doi":"arxiv-2409.08911","DOIUrl":"https://doi.org/arxiv-2409.08911","url":null,"abstract":"We explicitly describe the splitting of odd integral primes in the radical\u0000extension $mathbb{Q}(sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial\u0000in $mathbb{Z}[x]$. Our motivation is to classify common index divisors, the\u0000primes whose splitting prevents the existence of a power integral basis for the\u0000ring of integers of $mathbb{Q}(sqrt[n]{a})$. Among other results, we show\u0000that if $p$ is such a prime, even or otherwise, then $pmid n$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Debmalya Basak, Raghavendra N. Bhat, Anji Dong, Alexandru Zaharescu
The ternary Goldbach conjecture states that every odd number $m geqslant 7$ can be written as the sum of three primes. We construct a set of primes $mathbb{P}$ defined by an expanding system of admissible congruences such that almost all primes are not in $mathbb{P}$ and still, the ternary Goldbach conjecture holds true with primes restricted to $mathbb{P}$.
{"title":"Almost all primes are not needed in Ternary Goldbach","authors":"Debmalya Basak, Raghavendra N. Bhat, Anji Dong, Alexandru Zaharescu","doi":"arxiv-2409.08968","DOIUrl":"https://doi.org/arxiv-2409.08968","url":null,"abstract":"The ternary Goldbach conjecture states that every odd number $m geqslant 7$\u0000can be written as the sum of three primes. We construct a set of primes\u0000$mathbb{P}$ defined by an expanding system of admissible congruences such that\u0000almost all primes are not in $mathbb{P}$ and still, the ternary Goldbach\u0000conjecture holds true with primes restricted to $mathbb{P}$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Leonhard Eulertranslator, Jonathan David Evanstranslator
This is an English translation of Euler's 1750 paper "De numeris amicabilibus" (E152), the most substantial of his three works with this name. In it, he expounds at great length the ad hoc methods he has developed to search for pairs of amicable numbers, concluding with a list of around 60 new pairs.
{"title":"On Amicable Numbers","authors":"Leonhard Eulertranslator, Jonathan David Evanstranslator","doi":"arxiv-2409.08783","DOIUrl":"https://doi.org/arxiv-2409.08783","url":null,"abstract":"This is an English translation of Euler's 1750 paper \"De numeris\u0000amicabilibus\" (E152), the most substantial of his three works with this name.\u0000In it, he expounds at great length the ad hoc methods he has developed to\u0000search for pairs of amicable numbers, concluding with a list of around 60 new\u0000pairs.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a two-dimensional $mathbb F_p$-Selberg integral formula, in which the two-dimensional $mathbb F_p$-Selberg integral $bar S(a,b,c;l_1,l_2)$ depends on positive integer parameters $a,b,c$, $l_1,l_2$ and is an element of the finite field $mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.
{"title":"Notes on $2D$ $mathbb F_p$-Selberg integrals","authors":"Alexander Varchenko","doi":"arxiv-2409.08442","DOIUrl":"https://doi.org/arxiv-2409.08442","url":null,"abstract":"We prove a two-dimensional $mathbb F_p$-Selberg integral formula, in which\u0000the two-dimensional $mathbb F_p$-Selberg integral $bar S(a,b,c;l_1,l_2)$\u0000depends on positive integer parameters $a,b,c$, $l_1,l_2$ and is an element of\u0000the finite field $mathbb F_p$ with odd prime number $p$ of elements. The\u0000formula is motivated by the analogy between multidimensional hypergeometric\u0000solutions of the KZ equations and polynomial solutions of the same equations\u0000reduced modulo $p$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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