On the exponential diophantine equation Unx+Un+1x=Um
Let { U n } n ≥ 0 be the Lucas sequence. For integers x , n and m , we find all solutions to U n x + U n + 1 x = U m . The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication becaus...
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| Veröffentlicht in: | The Ramanujan journal 2024, Vol.64 (1), p.153-184 |
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| creator | Batte, Herbert Ddamulira, Mahadi Kasozi, Juma Luca, Florian |
| description | Let
{
U
n
}
n
≥
0
be the Lucas sequence. For integers
x
,
n
and
m
, we find all solutions to
U
n
x
+
U
n
+
1
x
=
U
m
. The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem
2.1
from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008). |
| doi_str_mv | 10.1007/s11139-023-00818-x |
| format | Article |
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{
U
n
}
n
≥
0
be the Lucas sequence. For integers
x
,
n
and
m
, we find all solutions to
U
n
x
+
U
n
+
1
x
=
U
m
. The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem
2.1
from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008).</description><identifier>ISSN: 1382-4090</identifier><identifier>EISSN: 1572-9303</identifier><identifier>DOI: 10.1007/s11139-023-00818-x</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Combinatorics ; Diophantine equation ; Field Theory and Polynomials ; Fourier Analysis ; Functions of a Complex Variable ; Logarithms ; Mathematics ; Mathematics and Statistics ; Number Theory</subject><ispartof>The Ramanujan journal, 2024, Vol.64 (1), p.153-184</ispartof><rights>The Author(s) 2024</rights><rights>The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-p142x-2bfada4ba326b0fd579c8b4b4cc8ee8d3fd02e6197060fd63a3136790057fed13</cites><orcidid>0000-0003-3882-0189 ; 0000-0002-4273-0066 ; 0000-0003-1321-4422 ; 0000-0002-0941-9604</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11139-023-00818-x$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11139-023-00818-x$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Batte, Herbert</creatorcontrib><creatorcontrib>Ddamulira, Mahadi</creatorcontrib><creatorcontrib>Kasozi, Juma</creatorcontrib><creatorcontrib>Luca, Florian</creatorcontrib><title>On the exponential diophantine equation Unx+Un+1x=Um</title><title>The Ramanujan journal</title><addtitle>Ramanujan J</addtitle><description>Let
{
U
n
}
n
≥
0
be the Lucas sequence. For integers
x
,
n
and
m
, we find all solutions to
U
n
x
+
U
n
+
1
x
=
U
m
. The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem
2.1
from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008).</description><subject>Combinatorics</subject><subject>Diophantine equation</subject><subject>Field Theory and Polynomials</subject><subject>Fourier Analysis</subject><subject>Functions of a Complex Variable</subject><subject>Logarithms</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><issn>1382-4090</issn><issn>1572-9303</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNpFkE1LxDAQhoMouK7-AU8Fj0t0JpM2zcGDLH7Bwl7sOaRt6nZZ027TQn--0RU8zcvMwwzzMHaLcI8A6iEgImkOgjhAjjmfz9gCUyW4JqDzmCkXXIKGS3YVwh4AJJBaMLn1ybhziZv7zjs_tvaQ1G3X72zMPvaPkx3bzieFn1eFX-H8WHxds4vGHoK7-atLVrw8f6zf-Gb7-r5-2vAepZi5KBtbW1laElkJTZ0qXeWlLGVV5c7lNTU1CJehVpDFcUaWkDKlAVLVuBppye5Oe_uhO04ujGbfTYOPJw2BjJ9J0iJSdKJCP7T-0w3_FIL50WNOekzUY371mJm-AW4AV5s</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Batte, Herbert</creator><creator>Ddamulira, Mahadi</creator><creator>Kasozi, Juma</creator><creator>Luca, Florian</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>C6C</scope><orcidid>https://orcid.org/0000-0003-3882-0189</orcidid><orcidid>https://orcid.org/0000-0002-4273-0066</orcidid><orcidid>https://orcid.org/0000-0003-1321-4422</orcidid><orcidid>https://orcid.org/0000-0002-0941-9604</orcidid></search><sort><creationdate>2024</creationdate><title>On the exponential diophantine equation Unx+Un+1x=Um</title><author>Batte, Herbert ; Ddamulira, Mahadi ; Kasozi, Juma ; Luca, Florian</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p142x-2bfada4ba326b0fd579c8b4b4cc8ee8d3fd02e6197060fd63a3136790057fed13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Combinatorics</topic><topic>Diophantine equation</topic><topic>Field Theory and Polynomials</topic><topic>Fourier Analysis</topic><topic>Functions of a Complex Variable</topic><topic>Logarithms</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Batte, Herbert</creatorcontrib><creatorcontrib>Ddamulira, Mahadi</creatorcontrib><creatorcontrib>Kasozi, Juma</creatorcontrib><creatorcontrib>Luca, Florian</creatorcontrib><collection>Springer Nature OA Free Journals</collection><jtitle>The Ramanujan journal</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Batte, Herbert</au><au>Ddamulira, Mahadi</au><au>Kasozi, Juma</au><au>Luca, Florian</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the exponential diophantine equation Unx+Un+1x=Um</atitle><jtitle>The Ramanujan journal</jtitle><stitle>Ramanujan J</stitle><date>2024</date><risdate>2024</risdate><volume>64</volume><issue>1</issue><spage>153</spage><epage>184</epage><pages>153-184</pages><issn>1382-4090</issn><eissn>1572-9303</eissn><abstract>Let
{
U
n
}
n
≥
0
be the Lucas sequence. For integers
x
,
n
and
m
, we find all solutions to
U
n
x
+
U
n
+
1
x
=
U
m
. The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem
2.1
from Mignotte (A kit on linear forms in three logarithms.
http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf
, 2008).</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s11139-023-00818-x</doi><tpages>32</tpages><orcidid>https://orcid.org/0000-0003-3882-0189</orcidid><orcidid>https://orcid.org/0000-0002-4273-0066</orcidid><orcidid>https://orcid.org/0000-0003-1321-4422</orcidid><orcidid>https://orcid.org/0000-0002-0941-9604</orcidid><oa>free_for_read</oa></addata></record> |
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| issn | 1382-4090 1572-9303 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Combinatorics Diophantine equation Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Logarithms Mathematics Mathematics and Statistics Number Theory |
| title | On the exponential diophantine equation Unx+Un+1x=Um |
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