Have I broken Fermat's Last Theorem?

Algebra Level 2

Find the smallest value of $a + b + c$ , given that $a^3 + b^3 = c^3$ , $a, b$ and $c$ are distinct integers and that $c$ is a non-negative integer.

The answer is 0.

2 solutions

Anish Puthuraya
Mar 25, 2014

Its obvious that no three positive integers can satisfy this equation, lest Fermat's Last Theorem would be violated.

Since the problem mentions ' $c$ ' as non-negative, note that it can also take the value $0$ . The smallest value will be achieved when this happens.

If $c=0$ , then,

$a^3+b^3 = 0$

Note that $a=n,b=-n$ (or vice-versa) satisfies this relation, with the advantage that $a+b=0$ (it helps achieve the minimum value)

Hence,
$a+b+c = n-n+0 = \boxed{0}$

As c is non- negative, then it can acquire 0 and therefore to achieve the smallest value of a+b+c, we do so. On putting, a^{ 3 } + b^{ 3 } = 0, we can have n and -n as values of 'a' and 'b' respectively. Therefore, the sum is 0. Or else, we can take all a,b and c as 0 to get the minimum sum

Rahul Bothra - 7 years, 1 month ago

a=1, b=-1, c=0

Amartya Anshuman - 7 years, 2 months ago

This is yet another example of where a math problem degenerates into legalism. If a, b, c are clearly supposed to be positive integers (excluding 0), and c > 0, then there exists no a, b, c that meets the condition. So why is the answer "0"? On the other hand, if 0 is not excluded, then we can have a = 1, b = 0, c= 1, so that the answer would be 2. An answer of "0" should not mean "there exists no solution", it should mean that a + b + c = 0, which is not possible if a, b, c, are all "positive integers" excluding 0.

As I have said before, and will keep saying it, mathematics is precise, but its Achilles' Heel is the way problems in it are worded. Than it is often not so precise any more, and becomes subject to arbitrary interpretations.

Michael Mendrin - 7 years, 1 month ago

Rishi Evans
Apr 29, 2014

since, c cannot be -ve, so for getting minimum value lets assume that c=0.....therefore a^3+b^3=0...=>a^3=-b^3....=>a=-b therefore,a+b+c=a+b+0=0...

Since c^3 is positive, there are 2 ways to satisfy the equation, either 1 of a^3 or b^3 negative or they both are negative

If one of them is negative, for example a^3, if we change the equation into b^3 = a^3+c^3 wouldnt it violates fermats last theorem?

If they are both negative then a^3+b^3+c^3=0

Which is impossible isnt it, since they are positive

Plz help me im confused

Joonathan Ryan - 4 years, 5 months ago

Have I broken Fermat's Last Theorem?

The answer is 0.

2 solutions

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