I Are there any other occurrences of these interesting equalities?

[Terry Coates]

A^p - B^p - C^p = A - B - C
With p > 1 this appears to occur only when p = 5: A = 17: B = 16 : C = 13

A^p - B^p - C^p = D^p - E^p - F^p = G^p - H^p - I^p = Y
A,B,C = 3,2,1
D,E,F = 9,8,7
G,H,I = 37,36,21
( Y = 64)

Are these really the only occurrences of these equalities?

 

[jbriggs444]

Are these really the only occurrences of these equalities?

You are asking about solutions where all variables take on positive integer values?

 

[mfb]

I'll assume A,B,C > 0, otherwise there are many trivial solutions.

p=2 up to A=20:

A   B   C
7   6   4
9   7   6
11   10   5
12   10   7
14   11   9
16   15   6
17   14   10
19   15   12

There are many solutions for larger A as well.

p=3 has many solutions as well, the smallest one is (16,15,9).

p=4, p=5, p=6 and p=7 don't have a solution for A<200 apart from the one you posted. Heuristic arguments suggest solutions are very rare.

 

[Terry Coates]

I mean all variables to be positive integers greater than zero.
Thanks for examples with p = 2 and 3, so let's change my question to have p > 3.
The case A,B,C = 17,16,13 represents the nearest to Fermat being wrong with p = 5

 

[Terry Coates]

Also the three sets of A,B,C for p = 4 represent the least possible value of Y (A, B & C all different from each other)

 

[Terry Coates]

And with p = 1,2 or 3 the least possible value of Y is zero with an infinite number of sets (Pythagoras triples when p = 2, 1 or 2 with p = 3)