Tom

Consider the equation

a³ + b³ = c³

has no integer solutions according to Fermat (and Andrew Wiles)

can be proven easily if b = a? 2a³ = c³ and (³√ 2)a = c but ³√ 2 is irrational completes the proof.

now take case b≠a; say b = na where b>a without loss of generality so n>1;

get ³√ (n³ +1) = c/a means that there is no integer n where ³√ (n³ +1) is rational.

My question is try to find a number n>1 so that b³ - a³ = c³ exists for integers solution.

³√ (n³ -1) = c/a ... can this be done?

I know that a cuboid has 8 vertices and 12 edges, so here's my problem:

I need to evaluate how many scaffold bars are needed to build a cuboid frame; regulations

require that all adjacent vertices pairs have a bar connecting them.

E.g. if the frame was a square ABCD then there would be bars from AB, BC, CD, DA, AC, BD

i.e. four edges and two diagonals.

I know answer is 28 but how!

I'd like to know if A,B,C member of Rn square matrix (assume n>1 fixed), do the following properties hold, if they do hold then just say YES, if Not then state why not by example or explanation -formal proofs are not reqd.

Assume in all cases, unless otherwise stated that matrix is non-singular i.e. det(A) =/= 0

I hope to extend the statements to Cn to see which ones fails - suggestions on this would be welcome -- keep it brief and to the point please!

(1) AB = BA [commutative for all A,B]

(2) AC = BC means A = B [uniqueness for all A,B]

(3) A(BC) = (AB)C [associative for all A,B,C]

(4) A(B+C) = AB +AC [distributive for all A,B,C]

(5) AB = B means A =I [unique identity, all B]

(6) AB = 0 means A =null [unique null, all B]

(7) AB =C then B =A^-1C [inverse for all A - except singular A]

(8) ABA^-1 = C then B ~ C [exists similarity transform, for all B]

(9) A^n for sny integer -n [exists powers for all A]

(10) det(AB) = det(A)det(B) im not sure!