I have an idea for a start.

Using: a³ + b³ + c³ - 1 is divisible by "p"

Perhaps only a³ + b³ can be divisible by p, so subtracting 1 requires adding 1, and that is the c³ term

Also:

a + b + c - 1 = p

( a³ + b³ + c³ - 1 ) / ( a + b + c - 1 ) = whole number

That is as far as I have got, and I have to go do something else now.