Can Fermat's Equation Be Solved with Non-Positive Integers?

[avery]

hi,

is there integers a, b, c that satisfy the equation an + bn = cn for n>2
(I don't mean positive integers)

thx.

 

[mathman]

I assume you mean aⁿ + bⁿ = cⁿ. Even if a, b, or c is negative, it won't make any difference. If n is even, it is obvious, while if n is odd you can replace the negative by its positive counterpart and rearrange terms so the equation is in its original form.

 

[Amir Livne]

There are many solutions where one of the numbers is 0: two examples are 6⁴+0⁴=6⁴ and 5³+(-5)³=0³.

There can be no solution with non-zero numbers, and this can be inferred from non-existance of positive solutions.
For even n, this is trivial: since aⁿ=(-a)ⁿ, a solution with negatives is also a solution with positive integers.
For odd n, (-a)ⁿ=-aⁿ, and you can always rearrange the terms to have a solution in positive numbers. This is a bit different for each assignment of signs to a, b, and c, but as an example, if a<0,b>0,c>0 and aⁿ+bⁿ=cⁿ, then cⁿ+(-a)ⁿ=bⁿ is a solution in positive numbers.