(In reply to
solution by Daniel)
I found one step in the prescription for finding solutions for a and b, a+b = r^2 to be sufficient but not necessary. Indeed, there are solutions not included by the prescription. Below, the first pair follows the prescription for a and b, while the second pair does not. I suspect there other derivations that yield more pairs, e.g. for r=12. Note - for the extra pairs, (a+b) is always a multiple of r.
If m is a factor of r, then some additional possible pairs for (a+b, a-b) are, respectively: 1: (m r^2, r/m); 2: (mr, r^2/m); 3: (r^3/m, m)
Some pairs will yield negative values for a and/or b.
All pairs must pass a parity test as well, for that division by 2. In particular, all the above methods will produce integers when r is odd. Pair 1 above will fail (by making non-integers) when r/m is odd, and Pair 4 will fail similarly when r is even and m is odd. Of course -a and/or -b will work just as well and the non-integer results (ending in 0.5) still yield integer cubes.
But the list of prescriptions is still incomplete. For example, in Pair 4, m need not be a factor of r, but of r^3 (e.g. r=6 and m=4). So there are further prescriptions for a and b.
( 21^2 - 15^2) = ( 15^2 - 3^2) = 6^3 = 216
( 78^2 - 66^2) = ( 42^2 - 6^2) = 12^3 = 1728
( 36^2 - 28^2) = ( 24^2 - 8^2) = 8^3 = 512
(210^2 - 190^2) = ( 90^2 - 10^2) = 20^3 = 8000
( 78^2 - 66^2) = ( 43^2 - 11^2) = 12^3 = 1728
( 6^2 - 3^2) = ( 14^2 - 13^2) = 3^3 = 27
( 10^2 - 6^2) = ( 17^2 - 15^2) = 4^3 = 64
( 55^2 - 45^2) = ( 35^2 - 15^2) = 10^3 = 1000
(120^2 - 105^2) = ( 60^2 - 15^2) = 15^3 = 3375
( 78^2 - 66^2) = ( 48^2 - 24^2) = 12^3 = 1728
( 21^2 - 15^2) = ( 29^2 - 25^2) = 6^3 = 216
(171^2 - 153^2) = ( 81^2 - 27^2) = 18^3 = 5832
(105^2 - 91^2) = ( 63^2 - 35^2) = 14^3 = 2744
( 78^2 - 66^2) = ( 57^2 - 39^2) = 12^3 = 1728
( 78^2 - 66^2) = ( 62^2 - 46^2) = 12^3 = 1728
(136^2 - 120^2) = ( 80^2 - 48^2) = 16^3 = 4096
(120^2 - 105^2) = ( 76^2 - 49^2) = 15^3 = 3375
( 21^2 - 15^2) = ( 55^2 - 53^2) = 6^3 = 216
(120^2 - 105^2) = ( 80^2 - 55^2) = 15^3 = 3375
( 15^2 - 10^2) = ( 63^2 - 62^2) = 5^3 = 125
( 36^2 - 28^2) = ( 66^2 - 62^2) = 8^3 = 512
(171^2 - 153^2) = ( 99^2 - 63^2) = 18^3 = 5832
Edited on September 16, 2018, 3:35 am
Edited on September 16, 2018, 3:35 am
Edited on September 17, 2018, 1:13 am
Edited on September 17, 2018, 5:29 pm
re: solution - more pairs
