I am a number theory newbie. So please explain all the steps.
Hello,
You can see that 2011 is a prime number. Thus its only possible factorisation is 1 2011. Why do I say that ? It's because we'll factorise a^3-b^3.
a^3-b^3=(a-b)(a^2+ab+b^2)
~~~~~~~~~~~~~~~~~~~~
Let's see what the signs of a & b are.
If a<0, then a^3<0 and so -b^3>0 that is to say b<0. And b^3>a^3 --> b>a --> b-a>0, thus a^2+ab+b^2<0, which is not possible because a²+b²>0 and a & b have the same sign and then ab>0.
Therefore a has to be positive.
------------------------------------
Let's assume that a & b > 0.
Then a²+ab+b² > 1. Thus the only possibility is that a²+ab+b²=2011 and a-b=1.
--> a=b+1. Substituting in a²+ab+b²=2011 :
b²+2b+1+b²+b+b²=2011.
3b²+3b=2010
b²+b=670
b²+b-670=0
=1+4*670=2681. But 2681 is not a perfect square. Thus b is not an integer. Which is not satisfying the conditions. Therefore, there are no a & b, positive integers, such that a^3-b^3=2011.
------------------------------------
What if a>0 and b<0 ?
Well, try to think about that
@CB : not sufficient, eh ?
Edit : my 27th ^^
Hello, fardeen_gen!
A slight variation of Moo's solution . . .
Prove that .a³ - b³ .= .2011 .has no integer solutions.
We have: .(a - b)(a² + ab + b²) .= .2011
Since 2011 is prime, the factors are .±1 and ±2011
We have: .a - b .= .±1 . → . a .= .b ± 1 . [1]
. . . .and: .a² + ab + b² .= .±2011 . [2]
Substitute [1] into [2]: .(b ± 1)² + b(b ± 1) + b² .= .±2011
This simplifies to two equations:
. . . . .b² + b - 670 .= .0
. . . . . . . . . . . . . . . . . . ... and neither has integer solutions.
. . 3b² - 3b + 2012 .= .0