Can you find positive integer triples , with and cubefree and

If , and

The condition and 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:

are all cubefree; , 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

……………………

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.

The condition

and are cubefree numbers

is not always satisfied

E.g. {10,10,999}; 999 is not a cubefree number

then