
Numbers of the form a^2  ab + b^2 and such
Posted:
Mar 24, 2014 2:33 AM


Hi, How to prove that for any odd prime p of the form 1 mod 3, we can always find two positive numbers a and b such that p^3 = a^2  ab + b^2 (assume (a,b) = 1)
In fact any number q^3 with prime factors of the form 1 mod 3 can be expressed as a^2  ab + b^2. I would like to know how to prove this and why there is this restriction of 1 mod 3.
In fact, if we write a^n + b^n = (a+b) Q(a,b,n), where n odd and > 2, then, we can always find a,b such that Q(a,b,3) is a cube.
Example: Q(1,19,3) = 7^3; Q(7793, 75473, 3) = (7*13*19)^3 and so on.
In fact Q(a,b,3) is interesting in that Q(a,b,3) = Q(b, ba, 3) and this is not true for any other n.
However, Q(a,b,n) where n > 3 is never a nth power. (It is obvious that proving this is as hard as proving FLT but this is a stronger restriction).
The other thing is that all factors Q(a,b,n>3) are in the form 1 mod n. Not sure why this restriction exists? We know Q(a,b,n) = 1 mod n but why this applies to every factor which is a more severe restriction?
Thanks

