√(a + b/c) = a √(b/c) — Part 2

 
 
Can you find positive integer triples   (a, b, c),   with   b   and   c   cubefree and   (b, c) = 1

\sqrt [3]{a + b/c} \; = \; a \: \sqrt [3]{b/c}

 
If   b = a,   and   c = a^3 - 1

The condition   b   and   c   are cubefree   is not always satisfied

 
Here are the first few examples,

 
b = c   cubefree numbers:
2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30

(2,2,27),   (3,3,26),   (4,4,63),   (5,5,124),   (7,7,342)
a = b = 9   —->   c = 728   is not a cubefree number
a = b = 10   —>   c = 999   is not a cubefree number
(11,11,1330),   (12,12,1727),   (13,13,2196),   (14,14,2743),   (15,15,3374),  
a = b = 17   —>   c = 4912   is not a cubefree number
a = b = 18   —>   c = 5831   is not a cubefree number
a = b = 19   —>   c = 6858   is not a cubefree number
(20,20,7999),   (21,21,9260),   (22,22,10647), (23,23,12166)
a = b = 25   —>   c = 15624   is not a cubefree number
(26,26,17575)
a = b = 28   —>   c = 21951   is not a cubefree number
(29,29,24388),   (30,30,26999)

 
The necessary condition:

a-1, \; a, \; a+1   are all cubefree;   b = a,   c = a^3 - 1   is a cubefree number

is not always satisfied

 
E.g.

(1,2,3),   (2,3,4),   (3,4,5),   (4,5,6),   (5,6,7)

(9,10,11)   —>   c = 999   is not a cubefree number

(10,11,12),   (11,12,13),   (12,13,14),   (13,14,15),

(17,18,19)   —>   c = 5831   is not a cubefree number

……………………

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

3 Responses to √(a + b/c) = a √(b/c) — Part 2

  1. Paul says:

    In this case a = b and c = ((a^3 * b) – b)/a
    Format {a, b, c}
    giving:-
    {2,2,7}
    {3,3,26}
    {5,5,124}
    {6,6,215}
    {7,7,342}
    {10,10,999}
    {11,11,1330}
    {13,13,2196}
    {14,14,2743}
    {15,15,3374}
    {17,17,4912}
    {19,19,6858}

    Paul.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s