On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2023-05-15 DOI:10.1090/mcom/3854

Sebastian Heintze, Robert Tichy, Ingrid Vukusic, Volker Ziegler

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We prove that there exist effectively computable constants <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=\"application/x-tex\">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N 0\"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">N_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b comma c element-of double-struck upper Z\"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">b,c\\in \\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b greater-than upper B\"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>></mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">b> B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the equation <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U Subscript n Baseline minus b Superscript m Baseline equals c\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>b</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">U_n - b^m = c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has at most two distinct solutions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis n comma m right-parenthesis element-of double-struck upper N squared\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(n,m)\\in \\mathbb {N}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to upper N 0\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n\\geq N_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m greater-than-or-equal-to 1\"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">m\\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, we apply our result to the special case of Tribonacci numbers given by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T 1 equals upper T 2 equals 1\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T_1= T_2=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T 3 equals 2\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T_3=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Subscript n Baseline equals upper T Subscript n minus 1 Baseline plus upper T Subscript n minus 2 Baseline plus upper T Subscript n minus 3\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T_{n}=T_{n-1}+T_{n-2}+T_{n-3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 4\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n\\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By means of the LLL-algorithm and continued fraction reduction we are able to prove <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N 0 equals 2\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N_0=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B equals e Superscript 438\"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>438</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B=e^{438}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The corresponding reduction algorithm is implemented in Sage.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mcom/3854","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}

引用次数: 1

Abstract

Let ( U n ) n ∈ N (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B B and N 0 N_0 such that for any b , c ∈ Z b,c\in \mathbb {Z} with b > B b> B the equation U n − b m = c U_n - b^m = c has at most two distinct solutions ( n , m ) ∈ N 2 (n,m)\in \mathbb {N}^2 with n ≥ N 0 n\geq N_0 and m ≥ 1 m\geq 1 . Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1 T_1= T_2=1 , T 3 = 2 T_3=2 and T n = T n − 1 + T n − 2 + T n − 3 T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n ≥ 4 n\geq 4 . By means of the LLL-algorithm and continued fraction reduction we are able to prove N 0 = 2 N_0=2 and B = e 438 B=e^{438} . The corresponding reduction algorithm is implemented in Sage.

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丢番图方程𝑈_{𝑛}-𝑏^{𝑚}=𝑐

让U (n) n∈{n的n (U_n)在\ mathbb {n}}成为一个固定recurrence线性序列):通过和一些技术restrictions integers杂志》()。我们证明,以至于有存在effectively computable constants B B和N 0 N_0如此那车上为B、c∈Z B、c和B在\ mathbb {Z} >B b>B《equation U n−B = c U_n - B ^ m = c已经在大多数二distinct解决方案2 (n, m)∈n (n, m)在\ mathbb {n ^ 2的n和n≥0 \ geq N_0和m≥1 \ geq 1。而且,我们专心论点特别Tribonacci数字赐予的凯斯》由T = T = 2 = 1 T_1 = T_2 = 1 , 3 = 2 T_3 = 2 T T T和n = n−1 + T + n−2 T n−3 T_ {} = T_ {n-1} T_{已经开始}+ T_ {n-3}为n≥4 \ geq 4。我们可以证明N =2 N_0=2和B=e = B=e。corresponding算法正在以Sage的方式实现。

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464