Fractional part of √5 and √10

 

frac \, ( \,\sqrt { \,5 \,} \,) \; = \; \sqrt { \,5 \,} \; - \; 2

frac \, ( \,\sqrt { \,10 \,} \,) \; = \; \sqrt { \,10 \,} \; - \; 3

 
 

Prove that for any positive integer   n

(1)   there exists an integer   A   such that

(\sqrt { \,5 \,} \; - \; 2)^n \; = \; \sqrt { \,A + 1 \,} \; - \; \sqrt { \,A \,}

(2)   there exists an integer   B   such that

(\sqrt { \,10 \,} \; - \; 3)^n \; = \;  \sqrt { \,B + 1 \,} \; - \; \sqrt { \,B \,}

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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