| A Note on Arithmetic Function $\sigma(n)$ |
| Received:February 07, 2005 Revised:April 29, 2005 |
| Key Words:
amicable number perfect number equation positive integer solution.
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| Fund Project:the National Natural Science Foundation of China (10271037; 10671051); the Natural Science Foundation of Zhejiang Province (M103060); the Foundation of Zhejiang Provincial Department of Education (20061069)and Foundation of Hangzhou Normal University (200 |
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| Abstract: |
| Two distinct positive integers $m$ and $n$ are called amicable if $\sigma(m)=\sigma(n)=m+n$, where $\sigma(n)=\sum_{d\mid{n}}d$. This paper proves that $f(x,y)$ is not part of an amicable pair, where $f(x,y)=x^{2^{x}}+y^{2^{x}},x>y\geq{1},(x,y)=1$, one of $x$ and $y$ is odd number, the other is even. Hence, equation $\sigma(f(x,y))+\sigma(z)=f(x,y)+z$ has no positive integer solutions. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2007.01.017 |
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