On the equation

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• 作者：Saranya G. Nair ; ^{saranya@math.iitb.ac.in" class="auth_mail" title="E-mail the corresponding author} ; T.N. Shorey ^{shorey@math.iitb.ac.in" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Factorial ; Primes ; Product of factorials
• 刊名：Indagationes Mathematicae
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：27
• 期：3
• 页码：634-642
• 全文大小：173 K

文摘

For a given positive integer n id="mmlsi2" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si2.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=8ceb649ef9a7ac54ed09f5da7e89b8e8" title="Click to view the MathML source">nn>n class="mathContainer hidden">n class="mathCode">th altimg="si2.gif" overflow="scroll">nth>n>n>n>, we consider positive integers n id="mmlsi3" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si3.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=bcd41fe9a68112f3ad1800bc2a191388" title="Click to view the MathML source">a₁,a₂…,a_tn>n class="mathContainer hidden">n class="mathCode">th altimg="si3.gif" overflow="scroll">an>1n>,an>2n>…,atth>n>n>n> such that n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si4.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=05a0eb5ffaf04238e0d56a2cbb7ac07b" title="Click to view the MathML source">a₁!a₂!⋯a_t!=n!n>n class="mathContainer hidden">n class="mathCode">th altimg="si4.gif" overflow="scroll">an>1n>!an>2n>!⋯at!=n!th>n>n>n>. Luca proved that n id="mmlsi5" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si5.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=e436efcb9f4078eb54116ccce06be43a" title="Click to view the MathML source">n−a₁=1n>n class="mathContainer hidden">n class="mathCode">th altimg="si5.gif" overflow="scroll">n−an>1n>=n>1n>th>n>n>n> if n id="mmlsi6" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si6.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=83d3de7a8e9d65e894a1da90ef329da6" title="Click to view the MathML source">abcn>n class="mathContainer hidden">n class="mathCode">th altimg="si6.gif" overflow="scroll">abcth>n>n>n> conjecture holds and n id="mmlsi2" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si2.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=8ceb649ef9a7ac54ed09f5da7e89b8e8" title="Click to view the MathML source">nn>n class="mathContainer hidden">n class="mathCode">th altimg="si2.gif" overflow="scroll">nth>n>n>n> is sufficiently large. Erdős, Bhat and Ramachandra gave unconditional upper bounds on n id="mmlsi8" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si8.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=c3dbf7c877d465b6cfee930c3414c86f" title="Click to view the MathML source">n−a₁n>n class="mathContainer hidden">n class="mathCode">th altimg="si8.gif" overflow="scroll">n−an>1n>th>n>n>n> which we improve in this paper. Further we solve the equation when n id="mmlsi9" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0019357715001147&_mathId=si9.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=84cbac29ce324c76d53e5efc1144ca29" title="Click to view the MathML source">P(n+1)≤79n>n class="mathContainer hidden">n class="mathCode">th altimg="si9.gif" overflow="scroll">P(n+n>1n>)≤n>79n>th>n>n>n> by using our estimate.