Find necessary and sufficient conditions for the positive integer triple(A,B,C)to satisfy:

(A^3+B^3)/(A^3+C^3)=(A+B)/(A+C)

start with the given equation
(a^3+b^3)/(a^3+c^3)=(a+b)/(a+c)
factor the numerator and denominator on the left side
[(a+b)(a^2-ab+b^2)]/[(a+c)(a^2-ac+c^2)]=(a+b)/(a+c)
cancel out the common terms on both sides
(a^2-ab+b^2)/(a^2-ac+c^2)=1
bring denominator over to the left side
a^2-ab+b^2=a^2-ac+c^2
shuffle equation
b^2-c^2=ab-ac
factor
(b+c)(b-c)=a(b-c)
this holds identically if b=c so that is one set of solutions.
If b!=c, then we can divide out the b-c terms on both sides to get
a=b+c
which is the other set of solutions.

Thus all solutions are given by
(b,b,a)
(b,c,b+c)

Posted by Daniel on 2013-08-25 18:03:26