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 5001147&_mathId=si2.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=8ceb649ef9a7ac54ed09f5da7e89b8e8" title="Click to view the MathML source">n$n$, we consider positive integers 5001147&_mathId=si3.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=bcd41fe9a68112f3ad1800bc2a191388" title="Click to view the MathML source">a₁,a₂…,a_t$a_1,a_2\dots ,a_t$ such that 5001147&_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!$a_1!a_2!\cdots a_t!=n!$. Luca proved that 5001147&_mathId=si5.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=e436efcb9f4078eb54116ccce06be43a" title="Click to view the MathML source">n−a₁=1$n-a_1=1$ if 5001147&_mathId=si6.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=83d3de7a8e9d65e894a1da90ef329da6" title="Click to view the MathML source">abc$abc$ conjecture holds and 5001147&_mathId=si2.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=8ceb649ef9a7ac54ed09f5da7e89b8e8" title="Click to view the MathML source">n$n$ is sufficiently large. Erdős, Bhat and Ramachandra gave unconditional upper bounds on 5001147&_mathId=si8.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=c3dbf7c877d465b6cfee930c3414c86f" title="Click to view the MathML source">n−a₁$n-a_1$ which we improve in this paper. Further we solve the equation when 5001147&_mathId=si9.gif&_user=111111111&_pii=S0019357715001147&_rdoc=1&_issn=00193577&md5=84cbac29ce324c76d53e5efc1144ca29" title="Click to view the MathML source">P(n+1)≤79$P(n+1)\le 79$ by using our estimate.