Diophantine equations concerning balancing and Lucas balancing numbers
In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers B n are given by the recurrence B n = 6 B n - 1 - B n - 2 with initial conditions B 0 = 0 , B 1 = 1 and its associated Lucas balancing numbers C n are given by the recurrence C n = 6 C n - 1 - C n...
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| Veröffentlicht in: | Archiv der Mathematik 2017, Vol.108 (1), p.29-43 |
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| creator | Dey, Pallab Kanti Rout, Sudhansu Sekhar |
| description | In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers
B
n
are given by the recurrence
B
n
=
6
B
n
-
1
-
B
n
-
2
with initial conditions
B
0
=
0
,
B
1
=
1
and its associated Lucas balancing numbers
C
n
are given by the recurrence
C
n
=
6
C
n
-
1
-
C
n
-
2
with initial conditions
C
0
=
1
,
C
1
=
3
. First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (
x
,
y
) for the Diophantine equation
2
x
2
+
1
=
3
b
y
m
for any even positive integers
b
and
m
with
m
>
2
, given in (Int J Number Theory 11:1259–1274,
2015
). Also we prove that the Diophantine equations
B
n
B
n
+
d
…
B
n
+
(
k
-
1
)
d
=
y
m
and
C
n
C
n
+
d
…
C
n
+
(
k
-
1
)
d
=
y
m
have no solution for any positive integers
n
,
d
,
k
,
y
, and
m
with
m
≥
2
,
y
≥
2
and gcd
(
n
,
d
)
=
1
. |
| doi_str_mv | 10.1007/s00013-016-0994-z |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1880795006</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1880795006</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-63aae4fc2f0a63682a4e3c82cd57882e4e4891b20edc264de463b4292d70586d3</originalsourceid><addsrcrecordid>eNp1kD1PwzAQhi0EEqXwA9giMRvOH3XsERUKSJVYQGKzHOdSUrVOaycD_fW4CkMXpjvdPe-d9BByy-CeAZQPCQCYoMAUBWMkPZyRCZMcqDZCn5NJXguqtfm6JFcprTPMdWkmZPHUdrtvF_o2YIH7wfVtF1Lhu-AxhjasisptXPDHzoW6WA7epZNZGLYVxnRNLhq3SXjzV6fkc_H8MX-ly_eXt_njknrBVE-VcA5l43kDTgmluZMovOa-npVac5QotWEVB6w9V7JGqUQlueF1CTOtajEld-PdXez2A6berrshhvzSMq2hNDMAlSk2Uj52KUVs7C62Wxd_LAN71GVHXTbrskdd9pAzfMykzIYVxpPL_4Z-AcgubcY</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1880795006</pqid></control><display><type>article</type><title>Diophantine equations concerning balancing and Lucas balancing numbers</title><source>SpringerNature Journals</source><creator>Dey, Pallab Kanti ; Rout, Sudhansu Sekhar</creator><creatorcontrib>Dey, Pallab Kanti ; Rout, Sudhansu Sekhar</creatorcontrib><description>In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers
B
n
are given by the recurrence
B
n
=
6
B
n
-
1
-
B
n
-
2
with initial conditions
B
0
=
0
,
B
1
=
1
and its associated Lucas balancing numbers
C
n
are given by the recurrence
C
n
=
6
C
n
-
1
-
C
n
-
2
with initial conditions
C
0
=
1
,
C
1
=
3
. First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (
x
,
y
) for the Diophantine equation
2
x
2
+
1
=
3
b
y
m
for any even positive integers
b
and
m
with
m
>
2
, given in (Int J Number Theory 11:1259–1274,
2015
). Also we prove that the Diophantine equations
B
n
B
n
+
d
…
B
n
+
(
k
-
1
)
d
=
y
m
and
C
n
C
n
+
d
…
C
n
+
(
k
-
1
)
d
=
y
m
have no solution for any positive integers
n
,
d
,
k
,
y
, and
m
with
m
≥
2
,
y
≥
2
and gcd
(
n
,
d
)
=
1
.</description><identifier>ISSN: 0003-889X</identifier><identifier>EISSN: 1420-8938</identifier><identifier>DOI: 10.1007/s00013-016-0994-z</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Diophantine equation ; Initial conditions ; Integers ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Number theory ; Numbers</subject><ispartof>Archiv der Mathematik, 2017, Vol.108 (1), p.29-43</ispartof><rights>Springer International Publishing 2016</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-63aae4fc2f0a63682a4e3c82cd57882e4e4891b20edc264de463b4292d70586d3</citedby><cites>FETCH-LOGICAL-c316t-63aae4fc2f0a63682a4e3c82cd57882e4e4891b20edc264de463b4292d70586d3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00013-016-0994-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00013-016-0994-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>315,781,785,27928,27929,41492,42561,51323</link.rule.ids></links><search><creatorcontrib>Dey, Pallab Kanti</creatorcontrib><creatorcontrib>Rout, Sudhansu Sekhar</creatorcontrib><title>Diophantine equations concerning balancing and Lucas balancing numbers</title><title>Archiv der Mathematik</title><addtitle>Arch. Math</addtitle><description>In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers
B
n
are given by the recurrence
B
n
=
6
B
n
-
1
-
B
n
-
2
with initial conditions
B
0
=
0
,
B
1
=
1
and its associated Lucas balancing numbers
C
n
are given by the recurrence
C
n
=
6
C
n
-
1
-
C
n
-
2
with initial conditions
C
0
=
1
,
C
1
=
3
. First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (
x
,
y
) for the Diophantine equation
2
x
2
+
1
=
3
b
y
m
for any even positive integers
b
and
m
with
m
>
2
, given in (Int J Number Theory 11:1259–1274,
2015
). Also we prove that the Diophantine equations
B
n
B
n
+
d
…
B
n
+
(
k
-
1
)
d
=
y
m
and
C
n
C
n
+
d
…
C
n
+
(
k
-
1
)
d
=
y
m
have no solution for any positive integers
n
,
d
,
k
,
y
, and
m
with
m
≥
2
,
y
≥
2
and gcd
(
n
,
d
)
=
1
.</description><subject>Diophantine equation</subject><subject>Initial conditions</subject><subject>Integers</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number theory</subject><subject>Numbers</subject><issn>0003-889X</issn><issn>1420-8938</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kD1PwzAQhi0EEqXwA9giMRvOH3XsERUKSJVYQGKzHOdSUrVOaycD_fW4CkMXpjvdPe-d9BByy-CeAZQPCQCYoMAUBWMkPZyRCZMcqDZCn5NJXguqtfm6JFcprTPMdWkmZPHUdrtvF_o2YIH7wfVtF1Lhu-AxhjasisptXPDHzoW6WA7epZNZGLYVxnRNLhq3SXjzV6fkc_H8MX-ly_eXt_njknrBVE-VcA5l43kDTgmluZMovOa-npVac5QotWEVB6w9V7JGqUQlueF1CTOtajEld-PdXez2A6berrshhvzSMq2hNDMAlSk2Uj52KUVs7C62Wxd_LAN71GVHXTbrskdd9pAzfMykzIYVxpPL_4Z-AcgubcY</recordid><startdate>2017</startdate><enddate>2017</enddate><creator>Dey, Pallab Kanti</creator><creator>Rout, Sudhansu Sekhar</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2017</creationdate><title>Diophantine equations concerning balancing and Lucas balancing numbers</title><author>Dey, Pallab Kanti ; Rout, Sudhansu Sekhar</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-63aae4fc2f0a63682a4e3c82cd57882e4e4891b20edc264de463b4292d70586d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Diophantine equation</topic><topic>Initial conditions</topic><topic>Integers</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number theory</topic><topic>Numbers</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dey, Pallab Kanti</creatorcontrib><creatorcontrib>Rout, Sudhansu Sekhar</creatorcontrib><collection>CrossRef</collection><jtitle>Archiv der Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dey, Pallab Kanti</au><au>Rout, Sudhansu Sekhar</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Diophantine equations concerning balancing and Lucas balancing numbers</atitle><jtitle>Archiv der Mathematik</jtitle><stitle>Arch. Math</stitle><date>2017</date><risdate>2017</risdate><volume>108</volume><issue>1</issue><spage>29</spage><epage>43</epage><pages>29-43</pages><issn>0003-889X</issn><eissn>1420-8938</eissn><abstract>In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers
B
n
are given by the recurrence
B
n
=
6
B
n
-
1
-
B
n
-
2
with initial conditions
B
0
=
0
,
B
1
=
1
and its associated Lucas balancing numbers
C
n
are given by the recurrence
C
n
=
6
C
n
-
1
-
C
n
-
2
with initial conditions
C
0
=
1
,
C
1
=
3
. First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (
x
,
y
) for the Diophantine equation
2
x
2
+
1
=
3
b
y
m
for any even positive integers
b
and
m
with
m
>
2
, given in (Int J Number Theory 11:1259–1274,
2015
). Also we prove that the Diophantine equations
B
n
B
n
+
d
…
B
n
+
(
k
-
1
)
d
=
y
m
and
C
n
C
n
+
d
…
C
n
+
(
k
-
1
)
d
=
y
m
have no solution for any positive integers
n
,
d
,
k
,
y
, and
m
with
m
≥
2
,
y
≥
2
and gcd
(
n
,
d
)
=
1
.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00013-016-0994-z</doi><tpages>15</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0003-889X |
| ispartof | Archiv der Mathematik, 2017, Vol.108 (1), p.29-43 |
| issn | 0003-889X 1420-8938 |
| language | eng |
| recordid | cdi_proquest_journals_1880795006 |
| source | SpringerNature Journals |
| subjects | Diophantine equation Initial conditions Integers Mathematical analysis Mathematics Mathematics and Statistics Number theory Numbers |
| title | Diophantine equations concerning balancing and Lucas balancing numbers |
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