*a*+

*b*=

*c*

Then for

*every integer and all integers as c*we can write,
(

*a*+*b*)^{n}=*c*^{n}
To prove

*“No three positive integers a, b, and c can satisfy the equation**a*^{n}+*b*^{n}=*c*^{n}for any integer value of n greater than two”,
let

*b*=*a*+*d*(*d*=*b*−*a*) and substitute:*c*

^{n}=

*a*

^{n}+ (

*a*+

*d*)

^{n}

Therefore c cannot be an integer since all positive integers for

*n greater than two*are written as*Z*^{n}= (*a*+*b*)^{n}= (*a*+*a*+*d*)^{n}, where Z =*a*+*b*=*a*+*a*+*d*and Z represents one of the set of all real integers ℤ.*For n=2*,

*c*

^{2}= (

*a*+

*b*)(

*a*+

*b*) = (

*e*+

*d*)(

*e*−

*d*)

*c*

^{2}=

*e*

^{2}−

*d*

^{2}

*e*

^{2}=

*c*

^{2}+

*d*

^{2}

*Mark K Malmros*

*Chimney Point, VT*

*11/29/2014*