Posted by m on Thursday, July 23, 2009 at 7:18am.
can you answer this question in a different and more logical way than this method below:
we will examine the sum of cubes of two numbers, A and B. Without losing generality, we will further assume that
A=2nX and
B=2n+kY
where
X is not divisible by 2
n is a positive integer and
k is a nonnegative integer.
A3+B3
=(A+B)(A2AB+B2)
=2n(X + 2kY) 22n(X2  2kXY + 22kY²)
=23n(X + 2kY) (X²  2kXY + 22kY²)
Thus A3+B3 has a factor 23n, but not 23n+1 since X is not divisible by 2.
Since 103n+1 requires a factor of 23n+1, we conclude that it is not possible that
103n+1=A3+B3
The question is prove that a number 10^(3n+1), where n is a positive integer, cannot be represented as the sum of two cubes of positive integers.
Thanks> please explain in full steps and in the most logical order
Thanks once again

math  MathMate, Thursday, July 23, 2009 at 8:09am
The proof does look clumsy. It is however a little easier to follow if the exponents are displayed correctly. See the original version here:
http://www.jiskha.com/display.cgi?id=1247714839 
math  Jon Zhang, Monday, August 17, 2009 at 5:57am
Dude, it's going to be due in like three days. Do it yourself.

math  Sean, Tuesday, August 18, 2009 at 6:41am
it's due in two days, lol
SY